Sums of correlated exponentials: two
types of Gaussian correlation structures
vorgelegt von
Diplom-Mathematiker
Anton Klymovskiy (Klimovsky)
aus Charkiw
von der Fakult¨
at II – Mathematik und Naturwissenschaften
der Technischen Universit¨
at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Dirk Ferus
Berichter: Prof. Dr. Anton Bovier
Berichter: Prof. Dr. Erwin Bolthausen
Tag der wissenschaftlichen Aussprache: 24. Juni 2008
Berlin 2008
D83
Abstract
In this thesis, we study the limiting behaviour of the large sums of strongly correlated exponen-
tials as the number of their summands and the effective dimension of the correlation structure
simultaneously tend to infinity. We consider two types of such sums which are generated by two
a priori very different Gaussian correlation structures. The first type is a sum of hierarchically
correlated random variables which is based on the partition function of Derrida’s generalised
random energy model (GREM) with external field. The second type is an infinitesimal sum
of genuinely non-hierarchically strongly correlated random variables which is based on the
partition function of the Sherrington-Kirkpatrick (SK) model with multidimensional spins. We
consider the asymptotic behaviour (the thermodynamic limit) of these two sums on a logarith-
mic scale (i.e., at the level of free energy) and also at a more refined level of their fluctuations
(i.e., at the level of weak limiting laws). Interestingly for the SK model with multidimensional
spins, we find traces of a hierarchical organisation in the thermodynamic limit. This supports
the conjectured in theoretical physics universal behaviour of the sums of such sort.
Concerning the SK model with multidimensional spins, we obtain the following results.
We prove upper and lower bounds on the free energy of this model in terms of variational
inequalities. The bounds are based on a multidimensional extension of the Parisi functional. We
generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra
involving the GREM-inspired processes and Ruelle’s probability cascades. For this purpose, an
abstract quenched large deviations principle of the G¨
artner-Ellis type is obtained. We derive
Talagrand’s representation of Guerra’s remainder term for the SK model with multidimensional
spins. The derivation is based on well-known properties of Ruelle’s probability cascades and
the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi
functional by establishing a link with a certain class of semi-linear partial differential equations.
We embed the problem of strict convexity of the Parisi functional in a more general setting
and prove the convexity in some particular cases which, however, do not cover the original
setup of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of
multidimensional Gaussian a priori distribution of spins using Talagrand’s methodology of a
priori estimates.
Concerning the GREM in the presence of uniform external field, we obtain the following
results. We compute the fluctuations of the ground state and of the partition function in the
thermodynamic limit for all admissible values of parameters. We find that the fluctuations are
described by a hierarchical structure which is obtained by a certain coarse-graining of the ini-
tial hierarchical structure of the GREM with external field. We provide an explicit formula for
the free energy of the model. We also derive some large deviation results providing an expres-
sion for the free energy in a class of models with Gaussian Hamiltonians and external field.
Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic
limit tend to have a certain optimal magnetisation, as prescribed by strength of external field
and by parameters of the GREM.
Zusammenfassung
Diese Dissertation behandelt das Grenzwertverhalten der großen Summen der stark korrelierten
Exponenziale, w¨
ahrend die Anzahl ihren Summanden und die effektive Dimension der Korre-
lationsstruktur gleichzeitig gegen Unendlichkeit gehen. Wir betrachten zwei Arten dieser Sum-
men, die durch die a priori sehr unterschiedliche Gauß’schen Korrelationsstrukturen erzeugt
werden. Die erste Art ist eine Summe der hierarchisch korrelierten Zufallsvariablen, die auf der
Zustandssumme von Derrida’s generalised random energy model (GREM) mit externem Feld
basiert. Die zweite Art ist eine infinitesimale Summe der echt nicht-hierarchisch stark korre-
lierten Zufallsvariablen, die auf der Zustandssumme vom Sherrington-Kirkpatrick (SK) Modell
mit mehrdimensionalen Spins basiert. Wir betrachten das asymptotische Verhalten (der thermo-
dynamische Limes) dieser Summen auf der logarithmischen Skala (d.h., auf dem Niveau der
freien Energie) und außerdem auf dem pr¨
aziseren Niveau ihrer Fluktuationen (d.h., auf dem Ni-
veau der schwachen Grenzwertverteilungen). Interessanterweise finden wir auch im SK Modell
mit mehrdimensionalen Spins Spuren der hierarchischen Organisation im thermodynamischen
Limes. Dies unterst¨
utzt das hypothetische universale Verhalten dieser Summen aus der theore-
tischen Physik.
Bez¨
uglich des SK Modells mit mehrdimensionalen Spins erzielen wir die folgenden Ergeb-
nisse. Wir beweisen die oberen und unteren Schranken f¨
ur die freie Energie mittels Variati-
onsungleichungen, die auf der mehrdimensionalen Verallgemeinerung des Parisi-Funktionals
basieren. Wir setzen das Vergleichsschema von Aizenman, Sims und Starr und das von Guerra
ein, die die GREM-inspirierten Prozesse und Ruell’schen Wahrscheinlichkeitskaskaden invol-
vieren. Hierf¨
ur beweisen wir ein abstraktes “quenched” Prinzip der großen Abweichungen von
G¨
artner-Ellis Art. Mittels der Eigenschaften der Ruelle’schen Wahrscheinlichkeitskaskaden und
des Bolthausen-Sznitman Koaleszents leiten wir die Darstellung von Talagrand des Restterms
von Guerra f¨
ur das SK Modell mit mehrdimensionalen Spins ab. Wir untersuchen die Eigen-
schaften des mehrdimensionalen Parisi-Funktionals, indem wir eine Verbindung mit einer Ka-
tegorie semi-linearer partieller Differentialgleichungen herstellen. Wir betten das Problem der
strengen Konvexit¨
at des Parisi-Funktionals in einen allgemeineren Kontext ein. Wir zeigen die
Konvexit¨
at in einigen F¨
allen, welche jedoch nicht die urspr¨
ungliche Formulierung von Tala-
grand umfassen. Schließlich beweisen wir die Parisi-Formel f¨
ur die lokale freie Energie im Fall
der mehrdimensionalen Gauß’schen a priori Verteilung der Spins mit der Methodologie der a
priori Absch¨
atzungen von Talagrand.
Bez¨
uglich des GREMs in Anwesenheit des uniformen externen Felds erzielen wir die fol-
genden Ergebnisse. Wir berechnen die Fluktuationen des Grundzustandes und der Zustandss-
umme im thermodynamischen Limes f¨
ur alle zul¨
assigen Werte der Parameter. Wir finden, dass
im thermodynamischen Limes die Fluktuationen durch eine hierarchische Struktur beschrie-
ben sind. Diese Struktur ist eine Grobk¨
ornung der urspr¨
unglichen hierarchischen Struktur des
GREMs mit externem Feld. Wir stellen eine explizite Formel f¨
ur die freie Energie des Modells
zur Verf¨
ugung. Wir leiten auch einige Resultate zu den großen Abweichungen, welche einen
Ausdruck f¨
ur die freie Energie einiger Modelle mit Gauß’schen Hamiltonians und externem
Feld erm¨
oglichen, her. Schließlich beweisen wir, dass die grobk¨
ornigen Teile des Systems, die
in der thermodynamischen Limes auftauchen, dazu neigen eine optimale Magnetisierung zu ha-
ben. Diese Magnetisierung ist durch die St¨
arke des externen Felds und durch die Parameter des
GREMs vorgeschrieben.
To my mother and the memory of my father
Contents
Introduction ................................................................. 1
1 Framework .............................................................. 7
1.1 Physical roots and vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Statistical mechanics of spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Basicobjects .................................................... 8
1.2.2 Thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 The elementary Gibbs variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.4 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Spin-glasses .......................................................... 13
1.3.1 Basic mathematical objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Quenched disorder and random processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Realistic spin-glass models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4 Thermodynamic limit in spin-glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Some mean-field Gaussian spin-glass models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 The Sherrington-Kirkpatrick model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 The random energy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 The generalised random energy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.4 Between the GREM and SK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.5 Other mean-field spin-glass models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Replica symmetry breaking picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.1 Overlap distribution function and replica symmetry breaking . . . . . . . . . . . . 22
1.5.2 Limiting Gibbs measures and pure states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.3 Higher-level objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.4 Ultrametricity.................................................... 26
1.5.5 Free energy in the SK model: the Parisi formula . . . . . . . . . . . . . . . . . . . . . . . 26
2 Often used tools and some existing results ................................... 29
2.1 Oftenusedingredients.................................................. 29
2.1.1 Interpolation..................................................... 29
2.1.2 Concentration of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.3 Superadditivity................................................... 32
2.2 Limitingobjects ....................................................... 32
2.2.1 The Poisson-Dirichlet process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
VIII Contents
2.2.2 The Ruelle probability cascades. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.3 The Bolthausen-Sznitman coalescent and random permutations . . . . . . . . . . 38
2.3 Some results on Gaussian mean-field spin-glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.1 TheGREM...................................................... 42
2.3.2 The SK model with multidimensional spins . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.3 The Aizenman-Sims-Starr comparison scheme for the SK model . . . . . . . . . 49
3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins 53
3.1 Introduction........................................................... 53
3.2 Some preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 Covariance structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.2 Concentration of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.3 Gaussian comparison inequalities for free energy-like functionals . . . . . . . . 61
3.3 Quenched G¨
artner-EllistypeLDP ........................................ 64
3.3.1 Quenched LDP upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2 Quenched LDP lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 The Aizenman-Sims-Starr comparison scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 Naive comparison scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.2 Free energy upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.3 Free energy lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Estimates of Guerra’s remainder term in the SK model with multidimensional
spins .................................................................... 81
4.1 Guerra’s comparison scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.1 Multidimensional Guerra’s scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.2 Localcomparison ................................................ 83
4.1.3 Free energy upper and lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.4 The filtered d-dimensional GREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.5 A computation of the remainder term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 The Parisi functional in terms of differential equations . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.1 The Parisi functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2.2 On strict convexity of the Parisi functional and its variational representation 95
4.2.3 Simultaneous diagonalisation scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Remainderestimates ................................................... 97
4.3.1 A sufficient condition for µk-terms to vanish . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.2 Upper bounds on ϕ(2): Guerra’s scheme revisited . . . . . . . . . . . . . . . . . . . . . . 98
4.3.3 Adjustment of the upper bounds on ϕ(2)..............................102
4.3.4 Talagrand’s a priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.5 Gronwall’s inequality and the Parisi formula . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 The SK model with multidimensional Gaussian spins .........................111
5.1 Introduction...........................................................111
5.2 Proof of the local Parisi formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.1 The case of positive increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.2 Simultaneous diagonalisation scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.3 The Crisanti-Sommers functional in 1-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Contents IX
5.2.4 Replica symmetric calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.5 The multidimensional Crisanti-Sommers functional . . . . . . . . . . . . . . . . . . . . 122
5.2.6 Talagrand’s a priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2.7 The local low temperature Parisi formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 The GREM in the presence of uniform external field ..........................125
6.1 Introduction...........................................................125
6.2 Partial partition functions, external fields and overlaps . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 The REM with external field revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3.1 Free energy and ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3.2 Fluctuations of the ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.3 Fluctuations of the partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4 The GREM with external field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4.1 Fluctuations of the ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4.2 Fluctuations of the partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.4.3 Formula for the free energy of the GREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7 Some open problems and outlook ...........................................155
A Appendix ................................................................157
Notation Index ...............................................................159
References ...................................................................163
Introduction
In this thesis, we study the limiting behaviour of the large sums of correlated exponentials
ZM:=
M
∑
i=1
exp(XM(i)) (0.1)
as the number of their summands Mtends to infinity (here {XM(i)}M
i=1is an array of correlated
random variables). We shall consider the sums with the summands that have the following two
characteristic properties: a wide range of orders of magnitude and significant high-dimensional
correlations. Not surprisingly, the sums of this type emerge in a variety of applied contexts,
e.g., in models of interacting complex systems being considered in physics, biology, computer
science, social sciences, finance, etc.
We should emphasise that, for the purposes of modelling of special magnetic materials called
spin glasses, study of such sums Zwas initiated and brought to very sophisticated (though
mathematically unrigorous) levels of analysis in theoretical physics, see, e.g., M´
ezard et al.
(1987). For the most recent accounts of mathematical research on such sums Z, we refer to
a thorough monograph by Talagrand (2003), very readable introductory text-book of Bovier
(2006), and comprehensive collection of review articles (Bolthausen & Bovier, 2007).
In general, the first characteristic feature mentioned above precludes the applicability of
such standard tools of probability theory as the law of large numbers and the central limit
theorem, since Zmay well be (and actually, as we shall see, in many cases is!) dominated by a
small number of exceptionally large summands. In these cases we are forced to perform more
detailed analysis of such rare events. For this purpose, we shall invoke some techniques of large
deviations and also, perhaps more importantly, extreme value theory.
Contrary to extensively studied situations with one-dimensional correlation structures (e.g.,
Markov processes, where the involved random variables are indexed by a real time variable),
we shall deal with the high-dimensional correlation structures, where effective dimension of the
index set grows with M.
In the so far rigorously analysed sums Zof the above sort, a new class of limiting objects
appears. Moreover, contrary to what might have been expected, the limiting objects possess an
important hierarchical ultrametric structure. This structure appears to be universal, modulo the
number of levels of the hierarchy.
In this thesis, we consider two a priori very different instances of such sums of correlated
exponentials. These sums are inspired by the following mean-field spin-glass models.
2 Introduction
1. The generalised random energy model with external field. The generalised random en-
ergy model (GREM) was proposed by Derrida (1985) as a model of a random energy land-
scape with a priori ultrametric correlation structure. To define the family of correlated ran-
dom variables of interest, we first fix the ultrametric structure. Given N∈
N
, consider the
standard discrete hypercube ΣN:={−1;1}N. It will play the role of the index set. Define
the (normalised) lexicographic overlap between the configurations σ(1),σ(2)∈ΣNas
qL(σ(1),σ(2)):=(0,σ(1)
16=σ(2)
1
1
Nmaxnk∈[1;N]∩
N
:[σ(1)]k= [σ(2)]ko,otherwise. (0.2)
We equip the index set with the lexicographic distance defined as
dL(σ(1),σ(2)):=1−qL(σ(1),σ(2)).(0.3)
This distance is obviously an ultrametric, that is, for all σ(1),σ(2),σ(3)∈ΣN, we have
dL(σ(1),σ(3))≤maxndL(σ(1),σ(2)),dL(σ(2),σ(3))o.(0.4)
Let GREMN:={GREMN(σ)}σ∈ΣNbe the Gaussian random process on the discrete hyper-
cube ΣNwith a covariance of the following form
E
hGREMN(σ(1))GREMN(σ(2))i=ρ(qL(σ(1),σ(2))),(0.5)
where ρ:[0;1]→[0;1]is the non-decreasing right-continuous function such that ρ(0) = 0
and ρ(1) = 1. Given h∈
R
+, consider the Gaussian process X:=XN:={XN(h,σ)}σ∈ΣN
defined as
XN(h,σ):=GREMN(σ)+ h
√N
N
∑
i=1
σi,σ∈ΣN.(0.6)
The second summand in (0.6) is called the external field. The parameter hrepresents the
strength of external field. The GREM with external field is based on the following sum of
M:=2Nhierarchically correlated exponentials
ZN(β):=∑
σ∈ΣN
exphβ√NXN(h,σ)i,(0.7)
where β>0 is the real parameter called the inverse temperature. The sum (0.7) is called
the partition function. Of course, if β≈0 the summands in (0.7) are on the same scale of
magnitude and, hence, the law of large numbers applies. As we shall see below, the situation
becomes radically different and the law of large numbers breaks down, if βincreases above
a certain h-dependent threshold β0(h).
2. The Sherrington-Kirkpatrick model with multidimensional spins. The Sherrington-
Kirkpatrick (SK) model was introduced by Sherrington & Kirkpatrick (1975). This model
of a mean-field spin-glass has long been one of the most enigmatic models of statistical me-
chanics. Mathematically it boils down to the sum of correlated exponentials with a genuine
Introduction 3
non-hierarchical correlation structure. The correlations in the SK model are induced by
the Hamming distance on the discrete hypercube ΣN. In this thesis, we study the following
generalisation of the SK model. Consider arbitrary (not necessarily discrete) subset Σ⊂
R
d
and the corresponding index set ΣN:=ΣN. We define the family of Gaussian processes
X:={X(σ)}σ∈ΣNas
X(σ) = XN(σ):=1
N
N
∑
i,j=1
gi,jhσi,σji,(0.8)
where G:={gi,j}N
i,j=1consists of i.i.d. standard normal random variables and, for x,y∈
R
d,
hx,yi:=∑d
u=1xuyuis the standard Euclidean scalar product. The SK model with multidi-
mensional spins is based on the following infinitesimal sum of correlated exponentials
ZN(β):=ZΣN
expβ√NX(σ)dµ⊗N(σ),(0.9)
where µ∈Mf(Σ)is some arbitrary (not necessarily uniform or discrete) finite a priori
measure. The sum (0.9) is called the partition function.
To investigate limiting behaviour of the sums (0.7) and (0.9), it is natural to ask the following
questions.
As a first approximation, we consider the limiting behaviour of ZN(β)on a logarithmic scale.
Consider the random variable
pN(β):=1
NlogZN(β),(0.10)
where ZN(β)is given either by (0.9) or by (0.7). Somewhat abusing physical terminology we
shall call (0.10) the free energy.
Question 0.0.1. Does pNconverge to some random variable p as N ↑+∞? If convergence
holds, then in what sense (i.e., in probability, almost surely or L1)? Finally, can we compute p?
Assume that we have computed p(β), for all β>β0. It is intuitively clear that being able to
compute the limit
lim
β↑+∞β−1p(β) =:m
in many cases means that
lim
N↑+∞N−1/2max
σ∈ΣN
XN(σ)=m.(0.11)
Thus, knowing p(β)for large βprovides very sharp information on extremes of the process
XNas N↑+∞. That is, the constant mnot only bounds the extremes, but gives also their exact
value. This is unusually good.
The second natural more refined approximation is to study the “fluctuations” (weak limiting
laws) of the sum ZN.
4 Introduction
Question 0.0.2. Does there exist some scaling sequence of constants {CN}∞
N=1and a nontrivial
random variable Z0so that the weak convergence CNZN
w
−−−→
N↑+∞Z0holds? Is the distribution of
the random variable Z0to some extent universal?
Question 0.0.2 is, of course, harder than Question 0.0.1, but it seems to be exactly the ques-
tion that is claimed to be positively answered in theoretical physics.
The main purpose of this thesis is to compute the free energy (Question 0.0.1) and the fluctu-
ations of the partition function (Question 0.0.2) in the N↑+∞limit for the GREM with external
field and the SK model with multidimensional spins. We find that in both models similar hierar-
chical ultrametric structures do emerge in the N↑+∞limit. For the GREM with external field
we are able to give a complete positive answer to both Questions 0.0.1, 0.0.2. For the general
case of the SK model with multidimensional spins we are only able to give upper and lower
bounds on the free energy. However, for the special case, where µ(cf. (0.9)) is a Gaussian
distribution, we compute the limiting free energy and give a complete answer to Question 0.0.1.
To be more specific, concerning the SK model with multidimensional spins we prove upper
and lower bounds on the free energy (0.10) in terms of variational inequalities involving the
corresponding multidimensional generalisation of the Parisi functional (Theorems 3.1.1, 3.1.2,
4.1.1, 4.1.2). For this purpose, we generalise and unify the Aizenman-Sims-Starr (AS2) and
Guerra’s schemes for the case of multidimensional spins, and employ a quenched large devia-
tions principle (LDP) which may be of independent interest (Theorems 3.3.1 and 3.3.2). Both
schemes are formulated in a unifying framework based on the same comparison functional.
The functional acts on Gaussian processes indexed by an extended configuration space as in
the original AS2scheme. As a by-product, we provide also a short derivation of the remainder
term in multidimensional Guerra’s scheme (Theorem 4.1.4) using well-known properties of the
Ruelle probability cascades (RPC) and the Bolthausen-Sznitman coalescent. This gives a clear
meaning to the remainder in terms of averages with respect to a measure changed disorder.
The change of measure is induced by a reweighting of the RPC using the exponentials of the
GREM-inspired process.
We study the properties of the multidimensional Parisi functional by establishing a link be-
tween the functional and a certain class of non-linear partial differential equations (PDEs),
see Propositions 4.2.1, 4.2.2 and Theorem 4.2.2. We extend the Parisi functional to a contin-
uous functional on a compact space (Theorems 4.2.1, 4.2.2). We show that the class of PDEs
corresponds to the Hamilton-Jacobi-Bellman (HJB) equations induced by a linear problem of
diffusion control (Proposition 4.2.4). Motivated by a problem posed by Talagrand (2006c), we
show the strict convexity of the local Parisi functional in some cases (Theorem 4.2.4).
We partially extend Talagrand’s methodology of estimating the remainder term to the mul-
tidimensional setting (Theorem 4.1.4, Proposition 4.3.1, Theorem 4.3.1). In the case of multi-
dimensional Gaussian a priori distribution of spins we prove the validity of the Parisi formula
(Theorem 5.1.1).
Concerning the GREM in the presence of uniform external field, we find the fluctuations
of the ground state and of the partition function in the thermodynamic limit for all admissible
values of the model parameters (Theorems 6.1.1, 2.3.1, 6.1.3). An explicit formula for the free
energy of this model is given (Theorem 6.1.4). We also derive some large deviation results
providing an expression for the free energy of a class of models with external field and Gaussian
Hamiltonians. (Theorem 6.2.1). We prove that the coarse-grained parts of the system tend to
Introduction 5
have a certain optimal magnetisation as prescribed by the strength of external field and by
parameters of the GREM.
The thesis is organised as follows (see Figure 0.1 for the chapter dependency graph).
1 2
3
6
45
7
Fig. 0.1. Chapter dependency graph
Chapter 1 contains a description of some modelling ideas of statistical mechanics of disor-
dered systems. We fix some basic notation and terminology, and introduce the models used
throughout the work (in particular, the random energy model (REM), the GREM and the SK
model).
Chapter 2 is a short account of mathematical results available in the literature which are either
directly used in the arguments of this thesis or give a suggestive background and useful ex-
amples of what could be achieved. In particular, we discuss in this chapter some interpolation
techniques for Gaussian random variables, comparison theorems, concentration of measure,
the Poisson-Dirichlet processes, Ruelle’s probability cascades and the Bolthausen-Sznitman
coalescents.
Chapter 3 is the first chapter of this thesis dealing with the SK model with multidimensional
spins. We extend the AS2scheme to this model. We record some basic properties of co-
variance structure of the process (0.8) and establish the relevant concentration of measure
results. The chapter contains the tools allowing to compare and interpolate between the free
energy-like functionals of two different Gaussian processes. We derive a quenched LDP of
the G¨
artner-Ellis type under measure concentration assumptions. We conclude the chapter
with the derivation (based on the AS2scheme) of upper and lower bounds on the free energy
in terms of the saddle point of a Parisi-like functional.
Chapter 4 employs the ideas of Guerra’s comparison scheme in order to obtain upper and lower
bounds on the free energy in the SK model with multidimensional spins. It contains also a
useful analytic representation of the remainder term which is used in the next chapter. We
also study some properties of the multidimensional Parisi functional such as differentiability
and convexity. The chapter is concluded by a partial extension of Talagrand’s remainder term
estimates to the case of the SK model with multidimensional spins.
Chapter 5 concerns the SK model with multidimensional Gaussian a priori distribution of spins.
The main result of this chapter is a derivation of the local Parisi formula for the SK model
with multidimensional spins.
Chapter 6 is devoted to study of the fluctuations of the ground state and of the partition function
for the REM and GREM with external field. We also provide an explicit formula for limiting
free energy of these models. We obtain some large deviation results providing an expression
for the free energy of a class of models with Gaussian Hamiltonians and external field.
6 Introduction
Chapter 7 lists some open problems related to the models studied in this thesis.
Appendix A contains a proof of the almost superadditivity of the local free energy in the SK
model with multidimensional spins. It is an application of the Gaussian comparison results
we derived in Chapter 3.
Acknowledgements
Foremost, I would like to express my sincere gratitude to my adviser Prof. Anton Bovier for
sharing his vision, erudition and understanding of the fascinating subject of disordered systems
with me. Not less important for me were his optimistic approach to resolution of the problems,
his skill to understand the subject of matter almost without any words from my side, and his
very stimulating habit of dissecting the problem at hand by asking extremely sharp questions.
I would like to thank Prof. Erwin Bolthausen for agreeing to be the coexaminer of this thesis.
I have the pleasure thank the members of the Research Groups on Probability Theory and
Mathematical Finance (Technische Universit¨
at Berlin), on Interacting Random Systems (Weier-
strass Institute for Analysis and Stochastics) and on Probability Theory (Universit¨
at Potsdam)
for giving me the opportunity to start to understand the beauty and diversity of probability the-
ory.
I am indebted to Prof. Mariya Shcherbina for giving me the initiation in research on disor-
dered systems and for sharing her vision with me. Besides, I would also like to thank my lyceum
and university lecturers and professors – the representatives of the Kharkiv mathematical school
– for giving me the start in modern mathematics.
I am grateful to Prof. Michel Talagrand and Prof. Dmitry Panchenko for important remarks.
I would like to warmly thank my colleagues Stefan Ankirchner, Andrej Depperschmidt,
Alexander Drewitz, Matthias an der Heiden, Ivan Izmestiev, Mesrop Janunts, Pierre-Yves Louis,
Martin Slowik, Michel Sortais, Cristian Vormoor, Wiebke Wittm¨
uß for greatly contributing to
the cheerful atmosphere I enjoyed both at and after work. I thank Elise Luckfiel for her help
with peculiarities of the German language.
This work would totally be unthinkable without the exceptionally wise guide in life and
science, who was and is for me my prematurely passed away father. Completion of this thesis
would simply be impossible without love and steady support of my mother and professional
psychological help of my sister.
Financial support of both the Deutsche Forschungsgemeinschaft (through the Research
Training Group “Stochastic Processes and Probabilistic Analysis”) and the Helmholz-Gemein-
schaft is gratefully acknowledged.
1
Framework
In this chapter, we fix the framework and notations we shall use in the present work. Besides,
we try to give some motivational background to the set of problems we shall deal with.
1.1 Physical roots and vision
Starting from the 1970’s, an increasing amount of research in statistical mechanics has been
targeted at modelling the behaviour of magnetic materials collectively called spin-glasses. This
led to a group of models usually referred to as disordered spin systems. In general, disordered
systems are complex systems with spatial inhomogeneity. On physical grounds, the typical rep-
resentatives of these systems are alloys, composite materials, porous media, polymers, to name
a few. As became clear in recent decades, advances in understanding of the models of disor-
dered systems made in theoretical physics allow for their successful application to systems well
beyond the usual scope of physics. For example, we can name the complex systems arising
in computer science (combinatorial optimisation, computational complexity, see Hartmann &
Weigt (2005); M´
ezard & Montanari (2007), information processing, see M´
ezard & Montanari
(2007); Nishimori (2001); Opper & Saad (2001)), cognitive science (Amit, 1989; Arbib, 2003;
Nishimori, 2001), finance (agent based modelling, see Challet et al. (2005); Coolen (2005); risk
modelling, see Bouchaud & Potters (2003)), social sciences (Contucci & Graffi, 2007; Durlauf,
1999). Moreover, the models of disordered systems, the vision and heuristics developed in the-
oretical physics are abstract enough to be viewed as a collection of very sharp statements about
basic purely mathematical objects. This explains the ever growing interest in bringing these
heuristics on rigorous mathematical grounds in order to understand the real domain of their
applicability.
Often the precise mechanisms of inhomogeneity in disordered systems are not known in
advance or have a very complicated description. A way to model such situation is to assume
that instead of a full deterministic description of the system we have some statistical data on the
distribution of inhomogeneities. In that case, the inhomogeneities can be modelled by random
processes. This randomness is usually referred to as a quenched disorder1. Disordered systems
are often studied in the framework of statistical mechanics which is also the viewpoint adopted
in the present thesis.
1The term “quenched” emphasises a certain analogy with materials obtained by heat treatment which is called in metallurgy
quenching. The method amounts to rapid cooling of a heated up sample of metal in order to preserve (or freeze) inhomo-
geneities in it. In contrast, annealing is a process of slow cooling which gives enough time for homogenising diffusion
processes.
8 1 Framework
The main objective of statistical mechanics according to Nishimori (2001) is “the clarifi-
cation of the macroscopic properties of many-body systems starting from the knowledge of
interactions between microscopic elements”. In turn, statistical mechanics of disordered sys-
tems is a “particularly difficult, but also particularly exciting, branch of the general subject, that
is devoted to the same problem in situations when the interactions between the components are
very irregular and inhomogeneous, and can only be described in terms of its statistical prop-
erties” (Bovier, 2006). For introductory information on disordered systems we refer to Bovier
(2006).
Highly nontrivial heuristics such as the replica symmetry breaking (RSB) have been de-
veloped in theoretical physics to explain the very complex behaviour of even the “simplest”
spin-glass models both in dynamics and equilibrium (see, e.g., the landmark monograph by
M´
ezard et al. (1987)). Interestingly, these “simple” models are actually very basic but very little
studied mathematical objects which starting from the 1980’s became the subject of increasingly
active mathematical research.
One of the main messages of physical theory of the hierarchical RSB is the following ambi-
tious but unrigorous universality hypothesis.
The main characteristic observables (order parameters) of huge disordered spin systems
with long range interactions often have ultrametric (tree) structure.
The domain of applicability of the above heuristic seem to be broader than that of the central
limit theorem (CLT). Despite considerable efforts in recent years, only rather basic predictions
of physical theories were proved rigorously for the most interesting mean-field spin-glass mod-
els. Mathematical research on spin-glasses is still concentrated on several most simple cases
with more or less ad hoc methods specific to the concrete instance. There is, however, a hope
that there exists some general, yet not discovered, unifying mathematical theory explaining all
predictions on spin-glasses made in theoretical physics and, especially, the universality hypoth-
esis.
Rigorous understanding of applicability domain of the universality hypothesis would be a
very useful result, since the tree structures are dramatically more tractable than the general
graphic structures. This allows one not only to get substantially deeper analytical insight, but
also to construct faster algorithms solving hard problems at hand.
1.2 Statistical mechanics of spin systems
In this section, we recall some basic objects, terminology and facts used in statistical mechanics
of spin systems.
1.2.1 Basic objects
Spin system consists of sites indexed by some finite index set Λ. Assume Σ⊂
R
d. To each site
we attach an independent copy of the configuration probability space (Σ,S,µ). In this thesis,
we shall consider the following configuration spaces: Σ={−1;1}, compact subsets of
R
dand,
finally, the whole
R
ditself. Compactness of the configuration space Σis usually assumed. In
Chapter 5, we shall, however, treat an example of a non-compact configuration space. The
1.2 Statistical mechanics of spin systems 9
elements σ∈Σare traditionally called spins2or just configurations. A configuration gives
complete microscopic description of the site. In applications, the reference measure (or a priori
distribution)µis often just uniform. The overall configuration space of the whole system is
often a product set ΣΛ:=ΣΛ. Denote µΛ:=µ⊗Λand also SΛ:=SΛ. Then (ΣΛ,SΛ,µΛ)is
the (product) a priori probability space.
Remark 1.2.1. Sometimes we shall deal with the a priori measures µ∈Mf(Σ), i.e., with the
measures without probabilistic normalisation. This will not change (modulo a constant shift)
the quantities of interest.
Having defined the configuration space, we would like to have some means to compare the
individual configurations. For that purpose we assume that we are given some (measurable) cost
functional on the configuration space H:ΣΛ→
R
which is called the Hamiltonian. We say that
the configuration σ(1)∈ΣΛis energetically more favourable than the configuration σ(2)∈ΣΛ,
if H(σ(1))>H(σ(2)).
Remark 1.2.2. Physical convention is that the configurations with smaller energies are more
favourable. Mathematically this is only a matter of considering the Hamiltonian ˜
HΛ:=−HΛ.
The inverse temperature is the real parameter β∈
R
+. A Gibbs measure (also called the
Boltzmann distribution) GΛ(β)∈M1(ΣΛ)is a measure satisfying
dGΛ(β)
dµΛ
=1
ZΛ(β)expβHΛ,(1.1)
where necessarily
ZΛ(β):=µΛexpβHΛ.(1.2)
The normalisation constant (1.2) is referred to as the partition function.
Remark 1.2.3. Since we do not assume the compactness of Σ, we have to assume the existence
of (1.2), for all β≥0. The partition function (1.2) is a very familiar object. It is just the Laplace
transform of the random variable HΛon the a priori probability space (ΣΛ,SΛ,µΛ). This
immediately implies that the partition function is increasing and also (by the H¨
older inequality)
logarithmically convex function of β. Moreover, if Σis a compact set, then ZΛ(β)is an analytic
function of β.
A mathematical pattern behind (1.1) is as follows. At inverse temperature β=0, or equiv-
alently, at infinite temperature T :=β−1= +∞, we obviously have GΛ(β) = µΛwhich is a
sound physical fact. A system which is extremely heated up almost ignores its internal ener-
getic barriers. Hence, due to the analyticity of the Gibbs weights (with respect to β), we have
in a (sufficiently small) neighbourhood of β=0 that irrespective of the structure of the Hamil-
tonian the system looks just so as there is no energetic barriers at all.
On the other extreme, if we consider the low temperature limit, i.e., let β↑+∞(equivalently
T↓+0) the system is increasingly exposed to energetic costs. That is, the Gibbs measure is
more and more concentrated around the energetically most favourable configurations which are
2This terminology is a reminiscent of the fact that initially spin systems were developed as microscopic models of magnetism,
where each site models an atom or molecule and a spin state is an orientation of its individual magnetic moment.
10 1 Framework
the maxima of the Hamiltonian HΛ. Finally, in the low temperature limit the system “freezes”
completely at the maxima of the Hamiltonian HΛ. That is, the support of the Gibbs measure
coincides with the set argmaxHΛwhich is called the set of ground states (Remark 1.2.2 can
make this terminology more transparent). More formally, assume (for simplicity) that Σis a
compact set, HΛ∈C(Σ), and cardargmaxHΛ<∞. If we denote the uniform measure on the
set argmaxHΛby U(argmaxHΛ)∈M1(ΣΛ), then it is easy to show that
kGΛ(β)−U(argmaxHΛ)kTV −−−→
β↑+∞0.(1.3)
It is exactly this pattern relating the Gibbs measure and the maxima of the Hamiltonian HΛ
which makes the Gibbs measures attractive in applications. Often the problems of finding the
extrema of functions depending on a large number of variables are computationally hard. Gibbs
measures provide the whole family of objects indexed by the real parameter β∈
R
+. This allows
to approach (both analytically and algorithmically) the hard problem at β= +∞gradually.
1.2.2 Thermodynamic quantities
The observables are the measurable functions of spin configurations O:ΣΛ→
R
3. An important
quantity is the thermodynamic potential defined as
P
Λ(β):=logZΛ(β).(1.4)
Remark 1.2.3 implies that the thermodynamic potential is convex. This quantity has the follow-
ing useful property (cf. (1.3))
lim
β↑+∞
1
βP
Λ(β) = max
σ∈ΣΛ
HΛ(σ).(1.5)
Another useful property of this quantity is that it allows to compute averages with respect to the
Gibbs measure of other observables included in the Hamiltonian. For example, we have
d
dβP
Λ(β) = GΛ(β)HΛ.
More generally, given some macroscopic observable O:ΣΛ→
R
, fix some real parameter h∈
R
and define, for σ∈ΣΛ, the modified Hamiltonian HΛ(h;·):ΣΛ→
R
as follows
HΛ(h;σ):=HΛ(σ)+hO(σ).
Then we have
d
dhP
Λ(β) = GΛ(β)[O].
3In other words, the observables are the random variables on the configuration space.
1.2 Statistical mechanics of spin systems 11
1.2.3 The elementary Gibbs variational principle
Another motivation for introducing the thermodynamic potential (and the Gibbs measure) is the
fact that they solve a natural variational problem.
The functional S:M1(Σ)2→
R
+is called the relative entropy (or a Kullback-Leibler diver-
gence) of ν1with respect to ν2, if the following holds
S(ν1|ν2):=(ν2hlogdν1
dν2i,ν1ν2,
+∞,otherwise.
(1.6)
Define the functional ϕΛ(β;·):M1(ΣΛ)→
R
:=
R
∪{−∞}as
ϕΛ(β;ν):=βν HΛ−S(νkµΛ).
The following proposition holds true.
Proposition 1.2.1 (elementary Gibbs variational principle).For any β∈
R
+, we have
P
Λ=ϕΛ(β;GΛ(β)) = max
ν∈M1(ΣΛ)ϕΛ(β;ν).(1.7)
Remark 1.2.4. See, e.g., Ellis (2006) for more complicated instances of the same proposition
for infinite-volume (cardΛ=∞) spin systems.
In words, (1.7) means that the Gibbs measure GΛ(β)realises the best possible balance be-
tween the relative entropy and energy. Besides, the Gibbs variational principle gives a straight-
forward way to obtain lower bounds on the thermodynamic potential by choosing various trial
measures in (1.7).
In Chapter 3, following the seminal work of Aizenman et al. (2003), we obtain another
variational scheme for the SK model with multidimensional spins. This scheme expresses the
free energy of this model as the value of the Parisi functional at its saddle point (Theorems 3.1.1
and 3.1.2).
1.2.4 Thermodynamic limit
It is often hopelessly hard to study exact behaviour of thermodynamic quantities of interest for
finite but large systems due to their enormous dimensionality and the presence of irregularities.
The receipt of statistical mechanics is to approximate the quantities of interest for the finite
systems by their counterparts from the limiting infinite system. This approximation procedure
is called the thermodynamic limit.
Limiting thermodynamic potential
Consider the sequence of finite index spaces {ΛN}N∈
N
and the corresponding family of the
Hamiltonians {HΛN}∞
N=1such that
lim
N↑+∞cardΛN= +∞.
In what follows, we assume that the index spaces {ΛN}N∈
N
are organised such that, for all
N∈
N
,ΛN⊂ΛN+1⊂Λ0, where Λ0is some countable set such that Λ0=SN∈
N
ΛN.
12 1 Framework
Question 1.2.1 (thermodynamic potential).What is the asymptotic behaviour of thermody-
namic quantities such as the thermodynamic potential P
ΛNin the thermodynamic limit, i.e.,
as N ↑+∞? Does the limit exists? What is the behaviour of the limit with respect to β?
To fix some natural scale, it is by convention assumed in statistical mechanics that thermo-
dynamic quantities grow asymptotically linearly4with size of the system, that is,
P
ΛN(β)∼
N↑+∞p(β)cardΛN,(1.8)
where p(β)is some real constant. Therefore, in view of Question 1.2.1 and for notational con-
venience, we define the density of thermodynamic potential which is called the free energy
as
pN(β):=1
cardΛN
P
ΛN(β).(1.9)
Note that the assumption (1.8) implies that
lim
N↑+∞pN(β) = p(β).(1.10)
Infinite-volume Gibbs measures
The following question is rather imprecise as it is stated.
Question 1.2.2 (limiting Gibbs measure).Can we describe the limiting behaviour of the se-
quence of Gibbs measures {GΛN(β)}N∈
N
? What are the limiting objects of this sequence?
A clean way to answer the previous question is based on the concept of the Dobrushin-
Lanford-Ruelle (DLR) states, see, e.g., Bovier (2006); Simon (1993). This approach requires,
however, more structure from the Hamiltonian than it has been assumed previously. In partic-
ular, the Hamiltonian should (1) be defined for any ΛbΛ0, (2) be the function defined on the
whole ΣΛ0, i.e., HΛ(·):ΣΛ0→
R
, and (3) have the following form
HΛ(σ) = ∑
A:A∩Λ6=/0
UA([σ]A),(1.11)
where, for AbΛ0, the interaction potential UA:ΣA→
R
is the measurable with respect to SA
function. In this case, given any index subset ΛbΛ0and any external condition σ(c) ∈ΣΛ0\Λ,
we define the local specification GΛ(β,σ(c))∈M1(ΣΛ0), for a measurable set ∆⊂ΣΛ0, as
GΛ(β,σ(c))(∆):=1
ZΛ(β,σ(c))ZΛ
1
∆(σqσ(c))expβHΛ(σqσ(c))dµΛ(σ),(1.12)
where
ZΛ(β,σ(c)):=ZΛexpβHΛ(σqσ(c))dµΛ(σ).
4In physics such quantities are called extensive.
1.3 Spin-glasses 13
The set G0(β)⊂M1(ΣΛ0)of all infinite-volume Gibbs measures (or states) is the set of
measures G0(β)∈G0(β)such that, for all index sets ΛbΛ0, and all external conditions
σ(c) ∈ΣΛ0\Λ, we have
G0(β){·|[σ]Λ0\Λ=σ(c)}=GΛ(β,σ(c)){·},(1.13)
G0(β)-almost surely. Equation (1.13) is called the DLR equation. Under certain regularity con-
ditions (see, e.g., (Bovier, 2006, Corollary 4.2.8)) it can be proved that the set of all infinite-
volume measures G0(β)is not empty and can be constructed by rather straightforward limiting
procedure. Namely, it can be constructed as a convex hull of a set of all weak limiting points (as
N↑+∞) for the following sequences of local specifications {GΛN(β,σ(c)
N)}N∈
N
with all possible
external conditions {σ(c)
N}N∈
N
⊂ΣΛ0\ΣΛN.
The above set of ideas and especially the condition (1.13) shows that the Gibbs measures
generated by the Hamiltonians (1.11) can be viewed as graphical (spatial) generalisations of
Markov chains (see, e.g., Preston (1976)). Such generalisations (e.g., Markov random fields)
are important in various applications such as information processing, see, e.g., Winkler (2003).
In this thesis, we shall deal with mean-field systems, where the external conditions are ir-
relevant. Moreover, the Hamiltonians of such systems are not representable in the form (1.11).
Thus, the described above framework of the DLR equations is not applicable in the context of
mean-field models. See Section 1.3.4, for a more precise formulation of the questions which
will be dealt with in the present work.
Remark 1.2.5 (infinite-volume Gibbs variational principle).An alternative to the DLR states
way of defining the infinite-volume Gibbs measure is by using the infinite-volume analogues of
Proposition 1.2.1, see, e.g., Ellis (2006).
Phase transitions
After passing to the thermodynamic limit, many kinds of interesting effects can occur at the level
of free energy (cf. (1.10)). For example, the free energy may become a non-analytic function of
β. Such situation is usually referred to as the phase transition. In the context of the DLR states,
another indicator of the phase transition at the level of the Gibbs measure is the situation, where
the set G0(β)contains more than one element.
1.3 Spin-glasses
In this section, we introduce disordered spin systems we are going to study in the present work.
These are the spin systems with strongly randomised interactions.
1.3.1 Basic mathematical objects
In this subsection, we introduce the deterministic objects that will be randomised afterwards
and used as building blocks of spin-glass models.
Deterministic instances
We start from definition of non-disordered Hamiltonians with pair interactions.
14 1 Framework
The interaction kernel is the real matrix A∈
R
Λ×Λ. The external field is the real vector
h= (hi)i∈Λ∈
R
Λ. The (pair interaction) Hamiltonian is the mapping HΛ(A,h):ΣΛ→
R
defined
as
HΛ(A,h;σ):=∑
i,j∈Λ
Ai,jhσi,σki+∑
i∈Λhhi,σii.(1.14)
Remark 1.3.1. More generally, for p ∈
N
, one can consider the p-spin interactions defined as
follows. Let A(p)∈
R
Λpbe a real tensor of rank p ∈
N
. We also set d =1for simplicity. Define
the p-spin Hamiltonian HΛ(A(p),h):ΣΛ→
R
in the following way
HΛ(A(p),h;σ):=∑
i1,...,ip∈Λ
Ai1,...,ipσi1σi2···σip+∑
i∈Λ
hiσi.(1.15)
Remark 1.3.2. The presence of external field (i.e., h 6=0) in (1.14) introduces a bias which
makes the mapping (1.14) less symmetric by excluding, e.g., the rotational symmetry
HΛ(A,0,Oσ) = HΛ(A,0,σ),
where O ∈O(d)is an arbitrary orthogonal matrix.
Remark 1.3.3. An important difference of the Hamiltonian (1.14) from the Hamiltonian (1.11)
is that the former one does not depend on spins outside Λ. For this reason, we say that the
former Hamiltonian does not interact with the exterior of Λ.
Graph of interacting sites
Note that the Hamiltonian (1.14) naturally induces the graph of interacting sites or interaction
graph G = (V,E), with vertices V:=Λand edges E:={(i,j)∈Λ2|Ai,j6=0}.
1.3.2 Quenched disorder and random processes
Disorder
In the sequel, we shall assume that the interaction matrix Ais the random matrix on the proba-
bility space of disorder (Ω,F,
P
). This probability space is always assumed to be large enough
to contain all random variables we shall be dealing with (except those explicitly defined on the
spin configuration space ΣΛ). We shall be interested in typical behaviour of systems governed
by the random Hamiltonians (1.14).
Random processes
Due to the presence of disorder, the spin-glass Hamiltonian (1.14) can be viewed as a random
process on the probability space of disorder that is indexed by the configuration space ΣΛ(cf.
Definition 2.1.1). This “Hamiltonian-as-a-random-process” point of view we shall adopt in what
follows. In the sequel, we shall consider not only the pair interaction random Hamiltonians but
also the “abstract” random processes HΛ:={HΛ(σ)}σ∈ΣΛN
on the probability space of disorder.
Such Hamiltonians induce the random free energy and the random Gibbs measure as specified
by (1.9) and (1.1).
To lighten the notation, we shall use the following simplifications. Given some random inter-
action matrix A, we shall write simply HΛ(h;σ)instead HΛ(A,h;σ). Given β∈
R
+and h∈
R
,
the random Gibbs measure induced by HΛ(h;σ)at inverse temperature βwill be denoted by
GΛ.
1.3 Spin-glasses 15
1.3.3 Realistic spin-glass models
We start from two motivating examples of the physical relevance.
The Ruderman-Kittel-Kasuya-Yoshida (RKKY) model
Physically, spin-glasses are amorphous magnetic alloys. We can imagine the following mathe-
matical model. Given some dimension d∈
N
, let XN⊂
R
dbe the box XN:= [−N;N]d. Suppose
we have a family of i.i.d. random variables {ri}Nd
i=1uniformly distributed on XN. Realisations
of the family rmodel the quenched positions of impurities in the sample of material XN. Thus,
we are dealing with quenched inhomogeneities5. Put ΛN:={1,...,Nd}. Let further J:
R
→
R
be an even function decaying to zero at infinity. Assume that the spin configuration space is the
unit sphere Σ:=
S
d−1. Given the direction of external field h ∈
R
d, we define the Hamiltonian
HRKKY
N(h):ΣΛN→
R
as
HRKKY
N(h;σ):=∑
i<j
i,j∈ΛN
J(ri−rj)hσi,σji+hh,∑
i∈Λ
σii,σ∈ΣΛN.(1.16)
A physical example of the function Jis the RKKY exchange potential
J(r):=1
krk3cos(krk).(1.17)
Physical reasons for this form of the exchange potential Jcan be found, e.g., in Fischer & Hertz
(1991). The Hamiltonian (1.16) becomes a rather complex expression with the above choice
of the exchange potential (1.17). The summands involving the exchange potential decay (albeit
slowly) as the distance between the interacting sites grows. Hence, the interaction between the
spins i,j∈ΛNwith large distance kri−rjkbetween them gives only a small contribution to
(1.16). Moreover, the exchange potential oscillates taking positive and negative values. Due
to these oscillations, there are summands in (1.16) with Ji,j:=J(ri−rj)>0 (ferromagnetic
interaction). In this case collinear spins σiand σjare energetically the most favourable. There
are, however, nonnegligible summands in (1.17) with Ji,j<0 (antiferromagnetic interaction).
In the latter case anticollinear spins are energetically the most favourable. These two cases may
impose contradicting requirements on the spin configuration σ, since, for the triple i,j,k∈ΛN
such that i<j<kwith non negligible probability we have Ji,j>0, Jj,k>0 and Ji,k<0. This
conflicting non transitive situation is called frustration.
Unfortunately, even on the heuristic qualitative level of theoretical physics the thermody-
namic limit of the RKKY model seems to be very hard to analyse, see, however, Zegarli´
nski
(1998).
The Edwards-Anderson (EA) model
The EA model (Edwards & Anderson, 1975) is an attempt to simplify the complicated picture
of the previous model, while retaining the most important features of it, namely the frustration,
quenched disorder and locality of interactions.
Let ΛN:=
Z
d∩[−N;N]dand also let Σ:={−1;1}⊂
R
. Let now {gi,j}i,j∈ΛNbe a family of
i.i.d. standard Gaussian random variables. Consider the following EA Hamiltonian
5An example of such material is the Cu-Mg alloy (with 1% of Mg).
16 1 Framework
HEA
N(h,σ):=∑
ki−jk2=1
i,j∈ΛN
gi,jσiσj+h∑
i∈Λ
σi.(1.18)
At the time of writing, there seems to be no consensus (except for the cases d=1 and
“d=∞”6) in the physical literature about the qualitative picture of the EA model in the thermo-
dynamic limit (see Newman & Stein (2007) for a recent review).
1.3.4 Thermodynamic limit in spin-glasses
To lighten the notation in what follows, we shall simply write Ninstead of ΛNin quantities
depending on ΛN. We shall sometimes drop the subscript index Nin occurrences, where it stays
fixed and its value is clear from the context.
Given the random interaction matrix A, we can naturally extend Questions 1.2.1 and 1.2.2 to
the case of quenched disorder and ask a few additional ones.
The extension of Question 1.2.1 to the case of quenched disorder is Question 0.0.1 from the
introduction. This is the first main question being addressed in the present work.
Question 1.3.1 (maximization algorithm for the average-case scenaio).Can one produce an
efficient (for the average-case scenario) algorithm for finding the maximiser of (1.14) or at
least an “almost maximiser” of it?
For algorithmic purposes of Question 1.3.1, it would be interesting to know an answer to the
quenched version of Question 1.2.2 and, in particular, to the following.
Question 1.3.2 (structure of the maximisers).What is the typical structure of the (almost) max-
imisers of (1.14)? How many maximisers are there? Are they very different?
Since the information about the almost maximisers can be gained from the geometry of the
Gibbs measure, it is natural to ask also the following.
Question 1.3.3 (structure of the random Gibbs measure).Does the random Gibbs measure look
almost surely uniform or it concentrates around certain random domains in the thermodynamic
limit? Can we construct the limiting random object that encodes the structure of the random
Gibbs measure in the thermodynamic limit?
Question 1.3.3 is certainly very informal. The partial excuse is that at the writing time there
is no general framework which gives a proper limiting description of the random Gibbs mea-
sure. See, however, Bovier (2006); Newman & Stein (2007), and references therein (cf. also
Section 1.5.3).
In this thesis, we shall instead of Question 1.3.3 deal with closely related Question 0.0.2
from the introduction.
To make Questions 1.3.2 and 1.3.3 more sensible, we have to specify some metric structure
on the configuration space. This metric structure depends, of course, crucially on the choice of
the Hamiltonian. In the next section we shall consider some representative examples.
6The latter case is the alias for the SK model on which more is said below.
1.4 Some mean-field Gaussian spin-glass models 17
1.4 Some mean-field Gaussian spin-glass models
In this section, we define the main models considered in the present work. They fall into the
class of mean-field spin-glass models which Hamiltonians are Gaussian processes indexed by
high dimensional configuration spaces.
The term mean-field refers to the idea to approximate complex interaction geometries (as in
the EA model) by certain effective, in some sense averaged, interactions with “simpler” geome-
tries. One way to make this approximation rigorous is Proposition 1.2.1, where one restricts the
maximisation to product measures, see, e.g., Opper & Saad (2001); Simon (1993).
1.4.1 The Sherrington-Kirkpatrick model
The celebrated Sherrington-Kirkpatrick model is based on a very simple instance of the Hamil-
tonian (1.14) on a fully connected graph G. The model ignores completely the locality of inter-
actions but retains high degrees of frustration and quenched disorder. A large body of heuristics
concerning spin-glasses were developed in theoretical physics during the analysis of the SK
model and then were applied to other models. In a sense, the SK model is the most understood
spin-glass model in theoretical physics. Some of heuristics concerning Question 0.0.1 have been
rigorously confirmed for the SK model in the series of papers by Aizenman et al. (2003); Guerra
(2003); Guerra & Toninelli (2002); Talagrand (2006b), see also Section 1.5.5 for some details.
But still, as of the time of writing, the model contains quite a few puzzles, especially concern-
ing Questions 1.3.1, 1.3.2 and certainly the hardest Question 1.3.3. For a recent short account
of mathematical results on this model see Bolthausen (2007) and references therein.
To define the SK model, we make the following assumption.
Assumption 1.4.1 (Ising spins).Let the index space be ΛN:= [1;N]∩
N
and let the spin config-
uration space be the Ising one, that is Σ:=ΣIsing :={−1;1}. Therefore, the spin configuration
space ΣΛNof the whole system is the discrete hypercube {−1;1}N. We also consider the uniform
a priori distribution µon the configuration space Σso that the individual spin configurations
are the Rademacher random variables on the a priori probability space.
Let {gi,j}N
i,j=1be a family of i.i.d. standard Gaussian random variables (the same as in
(1.18)). We set Ai,j:=1
√Ngi,jand hi:=h∈
R
. Hence, the Hamiltonian (1.14) assumes the
form of the SK Hamiltonian
HSK
N(σ):=1
√N
N
∑
i,j=1
gi,jσiσj+h∑
i∈Λ
σi.(1.19)
The reason for the normalisation factor 1/√Nin (1.19) is to satisfy our assumption (1.8).
Remark 1.4.1. The Gaussian character of the interaction matrix is not very crucial, at least, if
Question 0.0.1 is concerned, see Carmona & Hu (2006). However, it makes the analysis easier.
The SK Hamiltonian (1.19) is, thus, a Gaussian process on the spin configuration space ΣN.
For convenience, we shall also use the following centred Gaussian process
SKN(σ):=1
N
N
∑
i,j=1
gi,jσiσj.(1.20)
18 1 Framework
The distribution of this process is completely determined by its covariance structure
E
hSKN(σ(1))SKN(σ(2))i= 1
N
N
∑
i=1
σ(1)
iσ(2)
i!2
.
The overlap is the mapping R:Σ2
N→[−1;1]defined as
R(σ(1),σ(2)):=1
N
N
∑
i=1
σ(1)σ(2).(1.21)
In words, the quantity (1.21) measures the number of matching coordinates of the vectors
σ(1)and σ(2). In fact, if we denote the Hamming distance between σ(1)and σ(2)through
dH(σ(1),σ(2)), then we have
R(σ(1),σ(2)) = 1−2dH(σ(1),σ(2)).(1.22)
Remark 1.4.2 (canonical L2-distance).The Hamming distance can also be obtained (modulo
a constant factor) as the canonical L2-distance induced by the Gaussian process SKNon the
configuration space ΣN
dH(σ(1),σ(2)):=1
√2q
E
(SKN(σ(1))−SKN(σ(2)))2.
The model (1.20) can be seen as a particular instance of the models which have the covari-
ance structure of the following form
E
hXN(σ(1))XN(σ(2))i=f(R(σ(1),σ(2))),(1.23)
for some function f:[−1;1]→
R
is such that f(0) = 0 and f(1) = 1.
In particular, the Hamiltonian (1.15) with Ai1,...,ip:=N−p/2gi1,...,ip, where {gi1,...,ip}N
i1,...,ip=1
is the family of i.i.d. standard Gaussian random variables, has the covariance structure given by
(1.23) with f(q):=qp. This model is called Derrida’s p-spin model.
1.4.2 The random energy model
Perhaps, the simplest possible model in the framework of Section 1.3.2 is the random en-
ergy model. It was introduced by Derrida (1980, 1981), who considered the Gaussian process
{HREM(σ)}σ∈ΣNwithout any correlations. More formally, consider
HREM
N(σ):=√NREMN(σ),(1.24)
where {REM(σ)}σ∈ΣNis the family of 2Ni.i.d. standard Gaussian random variables. We work
under Assumption 1.4.1.
As opposed to pair interaction Hamiltonians (cf. (1.14)), the REM and, as well as its gen-
eralised version from the next section, lack a description in terms of interacting microscopic
spin variables σi. Nevertheless, while being a “toy model”, the REM serves quite surprisingly
as a building block of exact hierarchical structure arising in the thermodynamic limit of the SK
1.4 Some mean-field Gaussian spin-glass models 19
model (Section 2.2.1). Note that the SK model is a model with the genuine interacting spins
description (1.19). It is conjectured that the above mentioned hierarchical structure emerges
universally also for other mean-field spin-glass models (see Section 1.5.2). Related universality
facts on the level of random Hamiltonians (as opposed to the level of free energy and Gibbs mea-
sure, cf. Section 1.3.4) have been established in the series of papers by Bauke & Mertens (2004);
Ben Arous et al. (2006); Borgs et al. (2001, 2005a,b); Bovier & Kurkova (2004c, 2006a,b).
Remark 1.4.3. The covariance of the process REMN(·)can also be obtained as the limit of the
covariance of Derrida’s p-spin model as p ↑+∞.
Remark 1.4.4. The canonical L2-distance generated by the process REMN(·)induces, obvi-
ously, the discrete topology on ΣN.
It turns out that already this simple model without correlations demonstrates a phase transi-
tion. Indeed, the limiting free energy is (see Chapter 6) almost surely (and in Lp) given by
p(β) = (β2
2,β≤βc:=√2log2
β2
c
2+βc(β−βc),β>βc.(1.25)
The two cases in (1.25) can easily be explained by the breakdown of the law of large numbers
(LLN) in the partition function of the REM for β>βc.
The questions listed in Section 1.3.4 have been considered and settled for the REM by Bovier
et al. (2002). The above game of i.i.d. processes on the hypercube in the framework of Section
1.3.2 can also be played for non-Gaussian distributions, see Ben Arous et al. (2005).
1.4.3 The generalised random energy model
The generalised random energy model was proposed by Derrida (1985). A recent account of
mathematical results on the GREM can be found in Bovier & Kurkova (2007). The results
are substantially more elaborate than for the SK model. Even the hardest Question 1.3.3 can
be answered for this model completely (Bovier & Kurkova, 2003a). However, perhaps most
significantly, the importance of the GREM stems from the fact that it emerges in a variational
formula for the free energy of the a priori much harder SK model. The variational principle was
proposed by Aizenman et al. (2003) (see also Section 2.3.3). This principle allows, in particular,
to obtain upper bounds on the quenched free energy of the SK model. These bounds have been
proved to be exact in the thermodynamic limit (Talagrand, 2006b), as prescribed by the Parisi
formula.
The GREM is based on the following random Hamiltonian
HGREM
N(σ):=√NGREMN(σ),
where {GREMN(σ)}σ∈ΣNis the Gaussian random process satisfying (0.5). (The model uses
Assumption 1.4.1.)
Consider the space of discrete order parameters
Q0
n:={q:[0;1]→[0;1]|q(0) = 0,q(1) = 1,qis non-decreasing,
piece-wise constant with njumps}.(1.26)
20 1 Framework
Fig. 1.1. An example of discrete order parameter (overlap distribution function)
Recall the function ρfrom (0.5). Assume that ρ∈Q0
n. In what follows, we shall refer to ρas
the discrete order parameter. In this case, it is possible to construct the process GREMNas a
finite sum of independent Gaussian processes. Assume that
ρ(x) =
n
∑
k=1
qk
1
[xk;xk+1)(x),(1.27)
where
0=:x0<x1< ... < xn=1,(1.28)
0=:q0<q1< ... < qn=1.(1.29)
See Figure 1.1 for an example graph of discrete order parameter.
Let {ak}n
k=1⊂
R
be such that a2
k=qk−qk−1. We assume that, for all k∈[1;n]∩
N
, we have
xkN∈
N
7and also ak6=0. Denote ∆xl:=xl−xl−1.
Consider the family of i.i.d standard Gaussian random variables
{X(σ(1),σ(2),...,σ(k))|k∈[1;n]∩
N
,σ(1)∈Σx1N,...,σ(k)∈ΣxkN}.
Using these ingredients, for σ=σ(1)qσ(2)q...qσ(n)∈ΣN, we have
GREMN(σ)∼
n
∑
k=1
akX(σ(1),σ(2),...,σ(k)).(1.30)
Equivalence (1.30) is easily verified by computing the covariance of the right hand side. The
computation gives, for σ,τ∈ΣN
CovGREMN(σ)GREMN(τ)=qNqL(σ,τ).
Figure 1.2 shows the tree structure corresponding to the GREM with the piece-wise constant
function ρsuch that n=3. This leads to the tree with three “REM levels”.
7This condition is for notational simplicity. It means that we actually consider instead of Nthe increasing sequence
{Nα}α∈
N
⊂
N
such that Nα↑+∞as α↑+∞, satisfying Nαxk∈
N
, for all α∈
N
and all k∈[1;n]∩
N
.
1.4 Some mean-field Gaussian spin-glass models 21
q3=1
q2
q1
q0=0
2(x3−x2)Nbranches
2(x2−x1)Nbranches
2x1Nbranches
X(σ(1),σ(2))
X(σ(1))
X(σ(1),σ(2),σ(3))
Fig. 1.2. Structure of the GREM with the three levels of hierarchy
1.4.4 Between the GREM and SK model
We now review shortly the results on generalisations of the GREM available in the literature.
We classify these results according to the set of questions proposed above.
The GREM is generated by the lexicographic distance (0.3) on the discrete hypercube. Der-
rida & Gardner (1986) considered the GREM with “lexicographic” external field which is par-
ticularly well adapted to the natural lexicographic distance generated by the GREM Hamilto-
nian. In contrast to the work of Derrida & Gardner (1986), in Chapter 6, we consider the GREM
with the uniform (magnetic) field (0.6). This external field is the same as in the SK model, cf.
(1.19). We answer both Questions 0.0.1 and 0.0.2 for this model in Chapter 6.
Bolthausen & Kistler (2006) introduced an a priori non-hierarchical generalisation of the
GREM and analysed it on the level of Question 0.0.1. In a recent preprint, Bolthausen & Kistler
(2008) prove that the Gibbs measure of the non-hierarchical GREM hierarchically decomposes
in the thermodynamic limit which leads to ultrametricity (Section 1.5.4). This result corre-
sponds to the level of geometry of the Gibbs measure (Question 1.3.3). Due to its a priori
non-hierarchical character, the model of Bolthausen and Kistler is closer to the SK model than
the GREM.
The model of Bolthausen & Kistler (2006) was studied also by Jana (2007); Jana & Rao
(2006) at the level of Question 0.0.1 with a generalisation to non-Gaussian distributions of
22 1 Framework
the REM summands in (1.30). Moreover, Jana (2007) analysed the GREM with randomised
branching rates (1.28) at the level of Question 0.0.1.
1.4.5 Other mean-field spin-glass models
For the sake of completeness, we mention briefly some mean-field spin-glass models which are
not considered in this thesis. Among them are the neural network memory capacity models:
the perceptron and Hopfield models, see Bovier (2006); Shcherbina (2005); Talagrand (2003).
Important are also the models of neural network learning, e.g., Klimovsky (2005); Shcherbina
& Tirozzi (2002).
Very important for applications are diluted mean-field spin-glass models. These models are
based on the Hamiltonians with random interaction graphs (e.g., Erd˝
os-R´
enyi ones). A promi-
nent example is the satisfiability problem on random graphs M´
ezard & Montanari (2007); Ta-
lagrand (2003), random codes M´
ezard & Montanari (2007); Nishimori (2001). Naturally, other
combinatorial problems such as the assignment one can be formulated as mean-field spin-glass
models (Talagrand, 2003, Chapter 8). The main distinguishing feature of these models from the
ones considered in this thesis is that their Hamiltonians (1.19) are not Gaussian. The impor-
tant tool in this case seems to be the rigorous versions of the cavity method, see, e.g., Bovier
(2006); Talagrand (2003). The cavity method is also quite compatible with Gaussian process
techniques, as shown by Aizenman et al. (2003), see also Section 2.3.3 and Chapter 3.
1.5 Replica symmetry breaking picture
In this section, we shall sketch the heuristic picture of the thermodynamic limit in mean-field
spin-glasses developed in theoretical physics (Dotsenko, 2001; M´
ezard & Montanari, 2007;
M´
ezard et al., 1987; Nishimori, 2001; Parisi, 2007). This picture suggests how the list of ques-
tions stated in Section 1.3.4 should be answered.
In landmark series of papers, Parisi and collaborators developed the heuristic Ansatz called
the replica symmetry breaking, see M´
ezard et al. (1987) and references therein. This Ansatz very
plausibly explained effects occurring in a large class of disordered mean-field spin systems. We
shall shortly review this Ansatz in the following subsection.
1.5.1 Overlap distribution function and replica symmetry breaking
Assume that ΣN={−1;+1}N. The overlap between configurations σ(1),σ(2)∈ΣNis defined
as
RN(σ(1),σ(2)):=1
N∑
i∈ΛN
σ(1)
iσ(2)
i.
The overlap can be viewed as the random variable on the probability space (Σ2
N,S⊗2
N,G⊗2
N)(i.e.,
the overlap is an observable for such a two-times replicated system). To explain the subsequent
terminology, we note that one can think of the latter probability space as of two independent
replicas of the initial spin glass configuration space. However, the marginals of the random
measure G⊗2
Nare not independent random measures on the probability space of disorder, since
they depend on the same realisations of disorder.
Let us consider the (finite volume) overlap distribution function FN(β;·):[0;1]→[0;1]
defined as
1.5 Replica symmetry breaking picture 23
FN(β;q):=
E
hG⊗2
N(β)nRN(σ(1),σ(2))≤qoi.(1.31)
We note that the overlap distribution function is indeed a distribution function of some random
variable. We shall denote this random variable by q. Assume that the thermodynamic limit
F(β;q):=lim
N↑+∞FN(β;q)(1.32)
exists. The behaviour of the function (1.32) with respect to βis conjectured (M´
ezard et al.,
1987) to be as follows. There exists βc∈
R
+such that:
1. Replica symmetric (RS) regime: for β<βc, we have q∼δq(β), for some constant q(β)∈
R
.
2. RSB region: for β>βc, we have cardsuppq>1. Then either of the following two subcases
holds
a) Discrete RSB:
i. Single-step RSB (1-RSB): there exist some β(1)
c∈
R
and β(2)
c∈
R
+such that β(2)
c>
β(1)
c>βcand, for all β∈(β(1)
c;β(2)
c), we have
q∼w(β)δq1(β)+(1−w(β))δq2(β),
for some constants q1(β),q2(β)∈
R
and w(β)∈(0;1). This regime has been rigor-
ously proven to hold for the REM (Bovier et al., 2002) with β(2)
c= +∞, and also for
the p-spin model (Talagrand, 2000b, 2003), where β(2)
c<+∞. Traces of this regime
at the level of free energy (Question 0.0.1) were confirmed for the spherical model
by Talagrand (2006a). In this thesis, we show an analogous result for the SK model
with multidimensional Gaussian spins, see Chapter 5.
ii. Finite-step RSB (n-RSB): the 1-RSB regime is a particular case of the following sit-
uation
q∼
n+1
∑
k=1
wk(β)δqk(β),
for some real constants q1(β)< ... < qn+1(β)and some vector of weights w(β)∈
(0;1)n+1such that ∑n+1
k=1wk(β) = 1. See Figure 1.1 for the typical graph of the over-
lap distribution function. This regime has been rigorously proven for the GREM by
Bovier & Kurkova (2004a). In Chapter 6 we prove an analogous result for the GREM
with external field at the level of Question 0.0.2.
b) Continuous RSB or full RSB (FRSB): there exists β(f)
c>0 such that for β∈(β(f)
c;+∞),
there exist some deterministic weights wm,wM∈[0;1), deterministic constants qm,qM∈
R
, and a continuous random variable qcont with suppqcont = [qm(β);qM(β)] such that we
have the following decomposition
q∼wm(β)δqm(β)+wM(β)δqM(β)+(1−wm−wM)qcont.
This decomposition is believed to hold for the SK model and Derrida’s p-spin model for
large enough β(f)
c(M´
ezard et al., 1987). Probably the only (at the time of writing) rigor-
ous hint that this regime might hold for the SK model is obtained by Talagrand (2006b)
at the level of free energy (which in our classification corresponds to Question 0.0.1).
Bovier & Kurkova (2004b) have considered this regime for the continuous version of the
GREM (CREM) at the level of Question 1.3.3.
24 1 Framework
Remark 1.5.1. The behaviour of the random overlap distribution function
FN(β;q):=G⊗2
N(β)nRN(σ(1),σ(2))≤qo
can be very different from the behaviour of (1.31), see Parisi & Talagrand (2004).
1.5.2 Limiting Gibbs measures and pure states
The RSB described in the previous section is believed to have its roots in the structure of the
Gibbs measure.
In classical statistical mechanics of spin systems, Gibbs measures often form a convex com-
pact set G0(β), see, e.g., (Simon, 1993, Chapter III.5). The Krein-Milman theorem then assures
that there exists the set of extremal points corresponding to the set of all limiting Gibbs mea-
sures. These extremal points are called pure states.
Somewhat similar but substantially more complicated picture is conjectured (M´
ezard et al.,
1987) to be true for mean-field spin-glass systems in the thermodynamic limit. It is conjectured
that in the FRSB case the Gibbs measure decomposes into a countable convex combination
of “random pure states”, where the coefficients of the convex combination are also random
and independent of the pure states. Moreover, it is conjectured that the distribution of these
coefficients is universal and the coefficients are hierarchically clustered. See Section 2.2.2, for
the random weights which seem to suit this picture. The pure states are claimed (at least for the
SK model) to be completely characterised by the “limiting overlaps” which, in turn, are also
believed to possess hierarchical structure.
Unfortunately, at the time of writing there is no general mathematical definition of the con-
cept of limiting Gibbs measure for mean-field spin-glass systems which reflects the picture
predicted in theoretical physics. However, there are proper definitions and results in several
concrete cases, see the next subsection.
Remark 1.5.2 (metastates).The limiting objects called metastates comprehensively encode the
behaviour of the limiting Gibbs measures in the case of non-mean-field (lattice) disordered
systems (e.g., the EA model). See Bovier (2006); Newman & Stein (2007) and references therein
for precise definitions and discussion.
1.5.3 Higher-level objects
As emphasised, e.g., in (Ellis, 2006), study of higher-level objects such as empirical distribu-
tion functions and their large deviations is fruitful for understanding the properties of classical
(non-disordered) spin systems. This ideas are closely related to the Gibbs variational principle
(Proposition 1.2.1).
Large-deviations of empirical measures
We refer to Bolthausen (2007) for a suggestion of yet another higher-level object approach in
the spirit of large deviations of empirical measures in the context of the perceptron model. See,
e.g., Comets (1989) for some earlier results in this spirit.
Comparison schemes
Recent breakthroughs in understanding of mean-field spin-glasses can be viewed as rewards
on the quest for “proper” higher-level objects. These breakthroughs were initiated by the com-
parison schemes of Guerra (2003) and Aizenman et al. (2003). See Section 2.3.3 for a short
1.5 Replica symmetry breaking picture 25
review. See Chapters 3, 4 for an extension of the comparison approach to the SK model with
multidimensional spins.
Empirical overlap distribution functions
Bovier & Kurkova (2004a,b) have obtained a comprehensive description of the limiting Gibbs
measure for the REM, GREM and CREM using the empirical overlap distribution function.
To define this object, we need the following ingredients. Let Q0⊂[0;1][0;1]be the set of all
non-decreasing right-continuous piece-wise constant functions x:[0;1]→[0;1]with x(0) = 0,
x(1) = 1 such that xhas only a finite number of jumps (Figure 1.1). We will refer to the set Q0
as the set of discrete order parameters. We equip it with the L1topology. The compactification
of this space we denote by Qand call the set of order parameters. The latter space is obviously
isometric to the space of all non-decreasing c`
adl`
ag functions x:[0;1]→[0;1]such that x(0) = 0
and x(1) = 1 equipped with the L1topology.
Now suppose that the Gibbs measure GN(β)is induced by the GREM Hamiltonian. Consider
the “random overlap distribution field” xN(β;·):ΣN×[0;1]→[0;1]defined as
xN(β;σ(1),q):=GN(β){σ(2)∈ΣN:qL(σ(1),σ(2))≤q}.(1.33)
Note that given σ(1)∈ΣN, the above field (1.33) induces the random order parameter
xN(β;σ(1),·)∈Q0.
In words, (1.33) is the distribution of the overlap between given configuration σ(1)∈ΣNand the
“equilibrated” (i.e., Gibbs-distributed) configuration σ(2)∈ΣN. Finally, we define the empirical
overlap distribution KN(β)of M1(X0)as
M1([0;1]) 3KN(β):=ZΣN
δxN(σ(1),·)GN(β;dσ(1)).
The precise relation between the empirical object KN(β)and the overlap distribution function
(1.31) is the following
E
ZX
xKN(β;dx)(q) = FN(β;q).
It is shown by Bovier & Kurkova (2003b, 2004b) that
KN(β)w
−−−→
N↑+∞
K0(β),(1.34)
where K0(β)is the random element in M1(Q0). Moreover, the explicit distribution of the
random measure K0(β)is provided. In particular, this random measure K0(β)is the “countable
convex combination” (with random weights) of Dirac measures. In turn, the Dirac measures are
supported by the random overlap distributions corresponding to the atomic measures of the
following form
(1−w1)δ0+(w2−w1)δ¯x1+...+wl(β)δ¯xl(β),
26 1 Framework
where {wk}l(β)
k=1⊂[0;1]are certain random weights (0 ≤w1≤...≤wl(β)≤1) with hierarchical
organisation, {¯xk}l(β)
k=1⊂[0;1]is a certain coarse-graining of {xk}n
k=1, and l(β)≤n. This result
gives a precise mathematical meaning to the “convex-combination-of-the-pure-states” heuris-
tics mentioned in Section 1.5.2. Besides, this result comprehensively displays the hierarchical
“geometry” of the Gibbs measure in the thermodynamic limit. We refer to Bovier & Kurkova
(2007) for the extensive presentation. Essentially, the limiting empirical overlap distribution ob-
ject K0(β)is equivalent to the Bolthausen-Sznitman coalescent see Section 2.2.3 and references
therein.
Gibbs measure of the SK-like models
We refer to Talagrand (2000b, 2003) for the results on the structure of the Gibbs measure in the
thermodynamic limit for Derrida’s p-spin model (with large enough p) in the 1-RSB regime.
See also Talagrand (2007a) for some related results and conjectures for the SK model in the
FRSB regime.
1.5.4 Ultrametricity
In physical literature, it is believed that the canonical metric structure of the SK model weighted
by the Gibbs measure becomes ultrametric in the thermodynamic limit. One possible way to
formalise this (Talagrand, 2007b) is the following assertion about the three replicas which is
formulated quite in the spirit of (1.31) as
lim
N↑+∞
E
hG⊗3
N(β)hR(σ(1),σ(3))≥R(σ(1),σ(2))∧R(σ(2),σ(3))−εii=1 (1.35)
which is required to hold for each ε>0.
The (unproved) assertion (1.35) is believed to be caused by the hierarchical organisation of
the pure states.
Remark 1.5.3. The definition of ultrametric (0.4) implies that the metric space (Σ,dU)has the
following property. For any triangle with vertices σ(k), k ∈{1,2,3}, we have that the lengths
of least two of the three sides dU(σ(1),σ(2)),dU(σ(2),σ(3)), and dU(σ(1),σ(3))are the same.
Using the language of balls this means that, given two generic balls, the balls are either disjoint
or one of them contains the other.
The assertion (1.35) states roughly that the canonical L2distance generated by the SK model
weighted with the corresponding Gibbs measure is almost ultrametric in the thermodynamic
limit.
1.5.5 Free energy in the SK model: the Parisi formula
Using mathematically unrigorous heuristics of the “zero replica limit”, Parisi (see, e.g., M´
ezard
et al. (1987)) proposed a formula which answers Question 0.0.1 for the SK model. As we shall
see below, this formula seems to be in line with the general RSB picture. Hence, the Parisi
formula can be treated as an evidence of the RSB picture. This formula was rigorously proved
by Talagrand (2006b) for a class of models which essentially reduces to Derrida’s p-spin model
with even p≥2. This proof was extended by Panchenko (2005b) to cover the case which in
our nomenclature corresponds to the SK model with multidimensional spins with d=1 and
a priori measure with bounded support, cf. (0.9). See Section 2.3.2 for a short review of the
1.5 Replica symmetry breaking picture 27
available results in the case d>1, Chapter 3 for partial results in the case d>1, and Chapter 5
for extension to the case of multidimensional Gaussian spins.
Using the definitions of Section 1.5.3, we may state the following result.
Theorem 1.5.1 (the Parisi formula, Guerra (2003); Talagrand (2006b)).Given x ∈Q0, let the
function fx(q,y):[0;1]×
R
→
R
be a unique solution of the following backward semilinear
terminal value problem
(∂qf(q,y)+ 1
2∂2
yf(q,y)+x(q)(∂yf)2(q,y)=0,q∈[0;1),y∈
R
,
f(1,y) = logcoshβ√2(y+h),y∈
R
.(1.36)
Define the Parisi functional as
P(β,x):=fx(0,0)−β2
2Z1
0
qx(q)dq.
Then, for any β>0and any h ∈
R
, the following Parisi formula holds
p(β,h) = inf
x∈Q0
P(β,x),(1.37)
where p(β,h)is the limiting free energy of the SK model with the Hamiltonian (1.24).
Remark 1.5.4. It is easy to see (Guerra, 2003; Talagrand, 2006c) that the Parisi functional is
Lipschitz continuous with respect to x ∈Q0(recall that the space Q0is equipped with the L1
topology). Hence, it can be extended by continuity onto the whole compact space Q. Then the
extended Parisi functional attains its infimum on Q.
The Parisi formula (1.37) can more intuitively be reformulated in terms of the AS2scheme.
The scheme expresses the Parisi functional in terms of difference between the free energies of
two GREM-like models, see Section 2.3.3 for details. This shows explicitly the relevance of the
hierarchical structure of the GREM for the SK model.
In this thesis, we partially extend Theorem 1.5.1 to the case of the SK model with multidi-
mensional spins, see Chapters 3, 4 and 5.
2
Often used tools and some existing results
The present chapter consists of three parts. In Section 2.1, we record some tools and techniques
that are often used in the context of mean-field spin-glasses. In Section 2.2, we record some
known limiting objects and their basic properties that are extensively used in the subsequent
chapters. Finally, in Section 2.3, we give a short review of the known results on Gaussian mean-
field spin-glasses which are closely related to or are special cases of the results obtained in this
thesis.
2.1 Often used ingredients
In this section, we record for the future reference some basic results on Gaussian processes.
In what follows, whenever we speak about Gaussian random variables (or processes), we
assume by default that they are centred.
We also accept the following.
Definition 2.1.1 (random process).Let (Ω,F,
P
)be a probability space. Suppose also that
(S,S)is a measure space. Let I be an abstract index space. The collection of the measurable
mappings
X:={X(i):Ω→S}i∈I
is called the random process X.
In particular, we shall be interested in Gaussian random processes indexed by high-dimensional
index spaces. For a general discussion of Gaussian random processes we refer, e.g., to mono-
graphs by Adler (1990); Adler & Taylor (2007); Bogachev (1998); Dudley (1999); Fernique
(1997); Ledoux & Talagrand (1991); Lifshits (1995).
2.1.1 Interpolation
This subsection is devoted to a short account of comparison results between the functionals of
Gaussian processes.
Integration by parts
We begin by recalling well-known integration by parts formula which is the source of many
comparison results for functionals of Gaussian processes.
Let F:X→
R
be a functional on the linear space X. Given x∈Xand e∈X, the directional
(Gˆ
ateaux) derivative of Fat xalong the direction eis
30 2 Often used tools and some existing results
∂x eF(x):=∂tF(x+te)t=0.(2.1)
We now ready to state the following.
Lemma 2.1.1. Let {g(i)}i∈Ibe a real-valued Gaussian process (the set I is an arbitrary index
set), and let h be some Gaussian random variable. Define the vector e ∈
R
Ias e(i):=
E
[hg(i)],
i∈I. Let F :
R
I→
R
be such that, for all f ∈
R
I, the function
R
3t7→F(f+te)∈
R
(2.2)
is either locally absolute continuous or everywhere differentiable on
R
. Moreover, assume that
the random variables hF(g)and ∂g eF(g)are in L1.
Then
E
[hF(g)] =
E
[∂g eF(g)].(2.3)
The previous lemma coincides with (Panchenko, 2005b, Lemma 4) (modulo the differentia-
bility condition on (2.2) and the integrability assumptions which are needed, e.g., for (Bogachev,
1998, Theorem 5.1.2)).
Corollary 2.1.1 (integration by parts for finite dimensional Gaussian vectors).Using the nota-
tions of Lemma 2.1.1, suppose that cardI<∞. Assume, further, that the function F :
R
I→
R
has
the first derivatives of moderate growth.
Then
E
[hF(g)] = ∑
i∈I
E
[hgi]
E
∂iF(g).(2.4)
See (Talagrand, 2003, formula (A.41)).
Quadratic interpolation paths method
The following is a simple but powerful equality which is a consequence of the Gaussian inte-
gration by parts. It can be considered as a tool of comparison between two Gaussian processes
that are seen through the prism of expected value of some functional of them.
Proposition 2.1.1. Consider two independent Gaussian random vectors X ={Xi}n
i=1, Y =
{Yi}n
i=1and the n-variate function F :
R
n→
R
. We require that F has second-order derivatives
of moderate growth. Define
Z(t):=√tX +√1−tY.(2.5)
Then
d
dt
E
[F(Z(t))] = 1
2
n
∑
i,j=1
(
E
[XiXj]−
E
[YiYj])
E
∂2
i,jF(Z(t)).(2.6)
In particular,
E
[F(X)] =
E
[F(Y)]+ 1
2
n
∑
i,j=1Z1
0
(
E
[XiXj]−
E
[YiYj])
E
∂2
i,jF(Z(t))dt.(2.7)
2.1 Often used ingredients 31
See, e.g., the proof of (Talagrand, 2003, Proposition 2.4.3). Analogous ideas can be found
in Kahane (1986). An interesting generalisation of the previous proposition to more general
interpolation paths than (2.5) is (Aizenman et al., 2007, Corollary A.2).
Remark 2.1.1. Definition (2.5) and the fact that Z(0) = Y and Z(1) = X may explain the name
“quadratic interpolation”.
Corollary 2.1.2 (generalised Slepian’s lemma).Let X and Y be two independent d-dimensional
Gaussian vectors. Let D1and D2be some subsets of {1,...,d}×{1,...,d}. Assume that
E
[XiXj]≤
E
[YiYj],(i,j)∈D1
E
[XiXj]≥
E
[YiYj],(i,j)∈D1
E
[XiXj] =
E
[YiYj],(i,j)/∈(D1∪D2).
Let F :
R
d→
R
be the function with the second derivatives of moderate growth and such that
the following holds
(∂2
i,jF(x)≥0,(i,j)∈D1
∂2
i,jF(x)≤0,(i,j)∈D1.
Then
E
[F(X)] ≤E[F(Y)].(2.8)
See (Ledoux & Talagrand, 1991, Theorem 3.11). A similar result can be extracted from
Joag-Dev et al. (1983).
2.1.2 Concentration of measure
We refer to the monograph of Ledoux (2001) for an extensive account of results on concentra-
tion of measure. We state here a typical result of concentration of measure for i.i.d. Gaussian
random variables.
Theorem 2.1.1 (Gaussian concentration measure of bound).Equip
R
nwith the Euclidean norm.
Suppose F :
R
n→
R
is Lipschitzian with some constant L >0. Further, let {gi}n
i=1be a family
of i.i.d. standard normal random variables.
Then, for any t ≤0, we have
P
{|F(g)−
E
[F(g)]|≥t}≤2exp−t2
2L2.(2.9)
See, e.g., (Ledoux, 2001, Corollary 2.6) for a proof.
Proposition 2.1.2 (a measure conentration bound for the SK free energy).Using the notations
of Chapter 1, suppose Σ⊂
R
is a bounded set. Further, assume Ω⊂ΣN. Let {H(σ)}σ∈ΣNbe
the Gaussian process with the correlation structure satisfying the following condition
sup
σ(1),σ(2)∈ΣN
1
N
E
hH(σ(1))H(σ(2))i−ξ(R(σ(1),σ(2)))≤c(N),
32 2 Often used tools and some existing results
where ξ:
R
→
R
is some continuous function and c(N)−−−−→
N→+∞0. Define
X:=logZΩexpβHN(σ)dµN(σ).
Then, for any t ≥0, we have
P
{|X−
E
[X]|>2t√LN}≤2exp−t2,
where L :=max{ξ(σ2):σ∈Σ}+c(N).
See (Panchenko, 2005b, Lemma 12) for a proof.
2.1.3 Superadditivity
The following is a classical theorem which is usually attributed to Fekete with an extension due
to Aizenman et al. (2003).
Theorem 2.1.2 (superadditivity).Consider a real sequence {an}∞
n=1⊂
R
. Suppose it is super-
additive, i.e., for all n,m∈
N
, we have
an+m≥an+an.
Then there exists the following limit
lim
n→∞
an
n=:a0=sup
n∈
N
an
n∈
R
.
Moreover,
a0=inf
n∈
N
sup
n∈
N
an
n=lim
m→∞
1
mlim
n→∞lim
n→∞
E
[an+m−am].(2.10)
See, e.g., (Aizenman et al., 2007, Lemma B.1) for a proof.
Subadditivity-based arguments were historically used to prove the existence of free energy in
the thermodynamic limit, cf. Question 1.2.1. In the context of the SK model, the superadditivity
in conjunction with the quadratic interpolation method was used for the first time by Guerra &
Toninelli (2002).
2.2 Limiting objects
In this section, we shall shortly review several limiting objects that arise in the context of mean-
field spin-glass models with Gaussian Hamiltonians.
In Sections 2.2.1 and 2.2.2, we consider limiting higher-level objects for the exponentials
of the REM and GREM Hamiltonians, respectively. In Section 2.2.3, we consider a limiting
higher-level object for the GREM overlap structure.
2.2 Limiting objects 33
2.2.1 The Poisson-Dirichlet process
In this subsection, we shall deal with a limiting structure for the REM. It was first suggested by
Ruelle (1987). The limiting structure is closely related to the Poisson-Dirichlet process.
Using the notation of Pitman & Yor (1997), the object we shall deal with is the Poisson point
process PD(x,0). The generic two-parameter Poisson-Dirichlet processes PD(x,a)can also be
obtained within framework of the REM exponentials, see Talagrand (2007a).
Definition 2.2.1 (limiting REM exponentials, Ruelle (1987)).Assume x ∈(0;1). The limiting
REM exponentials ξ(x)is the Poisson point process on
R
+with the following intensity density
R
+3t7→xt−x−1.(2.11)
We refer, e.g., to Daley & Vere-Jones (2003); Kallenberg (1983); Leadbetter et al. (1983);
Resnick (1987) for the discussion of random measures and the Poisson point processes.
A limiting higher-level object
To motivate the Definition 2.2.1, we need the following linear scaling function
uN(x):=x
aN
+bN,(2.12)
where
aN:= (2Nlog2)1/2,
bN:=aN−√2
2aN
log(aN√2π).
Consider the rescaled REM process
REM(σ):=γu−1
N(REM(σ)),(2.13)
where γ:= (2log2)−1/2. The relation between the rescaled REM process (2.13) and the Poisson
point process ξ(x)is established through the convergence of the corresponding “higher-level ob-
ject” (cf. Section 1.5.3). Namely, define the empirical process of the REM exponentials WN(β)
– a random element in Mf(
R
+)– as
WN(β):=∑
σ∈ΣN
δexp[βREM(σ)].(2.14)
The following “low temperature behaviour” then holds.
Theorem 2.2.1 (the limit of the REM exponentials).Suppose β>√2log2. Let x(β):=
√2log2/β∈(0;1). The point process (2.14) converges to ξ(x(β)) in distribution. More for-
mally, we have
WN(β)w
−−−→
N↑+∞ξ(x(β)).
See, e.g., Bolthausen & Sznitman (2002); Bovier et al. (2002) for a proof.
By a slight abuse of notation, let {ξ(α)}α∈
N
be the enumeration of all atom positions for a
given realisation of ξon
R
+.
34 2 Often used tools and some existing results
Properties
The particular form of the density (2.11) implies the following result.
Theorem 2.2.2 (some basic properties of the REM exponentials).
1. For any ε>0, we have
E
[card{α∈
N
:ξ(α)>ε}] = ε−x.(2.15)
2. Almost surely, it is possible to reorder the atoms {ξ(α)}α∈
N
={ξ(i)}i∈
N
such that, for all
i∈
N
, we have
ξ(i)>ξ(i+1).
3. The atom positions have the following “profile”
ξ(i)i1/x−−−→
i↑+∞1.
4. The atoms have the following integrability property. The sum ∑∞
i=1ξ(i)vconverges if and
only if v >x.
5. The partition function Z :=∑∞
i=1ξ(i)is almost surely finite and has an infinitely divisible
distribution satisfying Z ∼2−1/x(Z+Z0), where Z0is an independent copy of Z.
6. The v-th moment of the partition function Z is finite, i.e.,
E
[Zv]<∞if and only if u <x.
For the proofs, see (Aizenman et al., 2007, Theorem 5.1).
Remark 2.2.1. Theorem 2.2.2 implies that the sequence {ξ(α)}α∈
N
can be treated as a random
element of Mf(
N
).
In the sequel, we shall assume that the random sequence {ξ(i)}i∈
N
is decreasing.
Random permutations and their distributions
To state the next property of the limiting REM we need some additional ingredients. Let Ybe
the non-negative random variable independent of ξsatisfying the following moment condition
λ(Y):=
E
[Yx]1/x<+∞.
Denote by νthe distribution of Y. Let, further, {Y(i)}i∈
N
be the i.i.d. copies of Y. Consider the
following multiplicative deformation of the limiting REM process
{ξ(i)}∞
i=17→{ξ(i)Y(i)}∞
i=1.(2.16)
Due to Theorem 2.2.2, the resulting sequence {ξ(i)Y(i)}i∈
N
can also be reordered into the
decreasing sequence {˜
ξ(i)}i∈
N
, i.e.,
˜
ξ(i)>˜
ξ(i+1),i∈
N
.
Let π:
N
→
N
be the corresponding random permutation, i.e., ˜
ξ(i) = ξ(π(i))Y(π(i)). Define
also ˜
Y(i):=Y(π(i)). Then the following result holds.
2.2 Limiting objects 35
Theorem 2.2.3 (the averaging property of the REM, the “quasi-stationarity”).The limiting REM
process satisfies
1. The averaging (or the “quasi-stationarity”) property:
∑
α∈
N
δ˜
ξ(α)∼∑
α∈
N
δλ(Y)ξ(α).(2.17)
2. The reordering effect: the sequence {˜
Y(i)}i∈
N
consists of i.i.d. random variables which are
independent of ˜
ξ. Moreover, the distribution of, e.g., ˜
Y(1)is absolute continuous with re-
spect to νand has the following density
R
+3˜
t7→ ˜
tx
λ(Y).
For a proof see (Ruzmaikina & Aizenman, 2005, Proposition 3.1) and (Aizenman et al.,
2007, Theorem 5.2).
Define the normalisation operation N:Mf(
N
)→M1(
N
)as follows
N(η){i}:=η{i}
η(
N
).(2.18)
Note, that Theorem 2.2.3 implies that
∑
α∈A
δN(ξ){α}∼∑
i∈
N
δN(˜
ξ){i}.
Ruzmaikina & Aizenman (2005) show that under some mild regularity conditions the “quasi-
stationarity” property (2.17) characterises the distribution of the point process ξ(x)completely.
The limiting Gibbs measure
The above results are precise enough to give an answer to Question 1.3.3 in the REM case.
It is shown by Bovier (2001); Bovier & Kurkova (2004a) (see also (Talagrand, 2003, Theo-
rem 1.2.1)) that, in particular, for β>√2log2, we have the following weak convergence
∑
σ∈ΣN
δGN(β){σ}−−−→
N↑+∞
N(ξ(x(β))),
where x(β):=√2log2/β.
For β∈[0;√2log2), the situation is substantially easier. Given an arbitrary M∈
N
and
η∈ΣM, we have
GN(β){σ∈ΣN:[σ]M=η}−−−→
N↑+∞
1
2M,
almost surely, see, e.g., (Bovier, 2006, Section 9.3) for more details.
36 2 Often used tools and some existing results
2.2.2 The Ruelle probability cascades
After the seminal work of Derrida (1985) who introduced the GREM, the corresponding limit-
ing probability structure – probability cascades – was identified by Ruelle (1987). The RPC is
the point process, which atoms possess certain hierarchical correlations.
More formally, we shall need the following index spaces. Given n∈
N
, define A0:=/0,
Ak:=
N
kand A:=An. The latter index space can be seen as an (infinitely wide) tree with
n-levels of hierarchy.
Define also the set of discrete order parameters with n jumps Q0
n⊂Q0which consists of the
functions of the following form
x(q) =
n
∑
i=0
xi
1
[qi;qi+1)(q),(2.19)
where, naturally,
0=:x0<x1< ... < xn<xn+1:=1,
0=:q0<q1< ... < qn<qn+1:=1,(2.20)
see Figure 1.1 for a sketch of x.
Definition 2.2.2 (Ruelle’s probability cascade, Ruelle (1987)).Given some x ∈Q0
n, let, for all
k∈[1;n]∩
N
and all α∈Ak, the point processes ξk,[α]k−1
be the independent ones with
ξk,[α]k−1∼PPP(
R
+3q7→xkq−xk−1∈
R
).(2.21)
The Ruelle probability cascade then is the point process ξ=ξ(x1,...,xn)with the atom posi-
tions {ξ(α)}α∈Andefined as follows
ξ(α):=
n
∏
k=1
ξk,[α]k−1(αk).(2.22)
A limiting higher-level object
Assume that the order parameter x∈Q0
nis such that, for all k= [1;n]∩
N
, the following holds
xk=k
n,
∆qk>∆qk+1.(2.23)
Define
γk:=∆qk
(2log2)∆xk1/2
.
To motivate Definition 2.2.2, consider the following rescaled GREM process
GREM(σ):=
n
∑
k=1
γku−1
∆xkN(X(σ(1),σ(2),...,σ(k))).(2.24)
2.2 Limiting objects 37
Define also the correspondent point process of the rescaled GREM exponentials
W(n)
N(β):=∑
σ∈ΣN
δexp[βGREM(σ)].(2.25)
The following theorem then describes the “very low temperature regime” of the limiting GREM.
Theorem 2.2.4 (the limit of the GREM exponentials, Bovier & Kurkova (2004a)).Put {βk:=
p2log2∆xk/∆qk}n
k=1⊂
R
+. Fix an arbitrary β>βn. For k ∈[1;n]∩
N
, define xk(β):=βk/β∈
(0;1).
Then the point process (2.25) converges to ξ(x1(β),...,xn(β)) in distribution. More for-
mally, we have
W(n)
N(β)w
−−−→
N↑+∞ξ(x1(β),...,xn(β)).
See (Bovier & Kurkova, 2004a, Theorem 1.7) for a proof of more elaborate result which
covers also all remaining intermediate regimes β∈[0;βn].
Remark 2.2.2. Condition (2.23) implies, in particular, that all jumps of x are the extreme points
of the their concave hull. See Theorem 2.3.2 for similar results in the case of generic order
parameters.
Properties
Theorem 2.2.5 (some elementary properties of the RPC).
1. The partition function Z :=∑α∈Anξ(α)is almost surely finite and, moreover,
Z∼Z(x1)
n
∏
k=2
E
Z(xk)xn−11/xn−1.
In particular, we have
E
[logZ]<∞.
2. Almost surely, it is possible to reorder the atoms {ξ(α)}α∈
N
={ξ(i)}i∈
N
such that, for all
i∈
N
, we have
ξ(i)>ξ(i+1).
That is, there exists the corresponding random permutation π:
N
→Anwhich satisfies
ξ(k) = ξ(π(k)).
See, e.g., (Aizenman et al., 2007, Theorem 5.3) for a proof.
The limiting Gibbs measure
As already mentioned above, Question 1.3.3 can completely be answered for the GREM. It is
shown by (Bovier & Kurkova, 2004a, Theorem 1.9) that, in particular, for β>βn, we have
∑
σ∈ΣN
δGN(β){σ}
w
−−−→
N↑+∞
N(ξ(xn(β))),
where ξ(xn(β)) ∼PD(xn(β),0). Moreover, at the level of limiting structures the normalisation
(2.18) of the RPC is equidistributed with the normalisation of the Poisson-Dirichlet process.
More precisely,
38 2 Often used tools and some existing results
N(ξ(x1,...,xn)) ∼N(ξ(xn)),
where ξ(xn)∼PD(xn,0). See, e.g., (Bolthausen & Sznitman, 1998, Lemma 2.1) for a proof.
Hence, interestingly, the hierarchical RPC structure flattens down to the Poisson-Dirichlet struc-
ture, if the point process of limiting Gibbs weights is concerned.
The hierarchical structure is, however, important for encoding the RSB picture (cf. Sec-
tion 1.5). It is this structure which is captured by the limiting object introduced in the following
section.
2.2.3 The Bolthausen-Sznitman coalescent and random permutations
A deep insight of Bolthausen & Sznitman (1998) was to identify the dynamical system gen-
erated by the RPC with remarkable properties which allow to perform many spin-glass cal-
culations in a more transparent way. In what follows, we shall use this dynamical system in
computations involving the replicas of the system (see Chapter 4).
A contribution of Bovier & Kurkova (2004a,b) was, in particular, to relate the RPC, the
continuous time coalescent process of Bolthausen & Sznitman (1998), and its dual – the contin-
uous time branching processes of Bertoin & Le Gall (2000) – with the original setup of Derrida
(1985).
Definition
Let ξ=ξ(x1,...,xn)be the RPC process. Theorem 2.2.5 guarantees that there exists the re-
arrangement ξ={ξ(i)}i∈
N
of ξ’s atoms in a decreasing order. Recall (0.2) and define the
(random) limiting ultrametric overlap qL:
N
2→[0;n]∩
Z
as
qL(i,j):=max{k∈[0;n]∩
Z
:[π(i)]k= [π(j)]k},(2.26)
where we use the convention that max /0 =0. This overlap valuation (2.26) induces the sequence
of random partitions of
N
into equivalence classes. Namely, given a k∈
N
∩[0;n], we define,
for any i,j∈
N
, the Bolthausen-Sznitman equivalence relation as follows
i∼
kjdef
⇐⇒qL(i,j)≥k.(2.27)
Note the conceptual similarity between the equivalence (2.27) and the limiting empirical
overlap distribution K0(β)from (1.34).
In what follows, we shall, by a slight abuse of the language, not distinguish between the
concepts of the equivalence relation and the partition of a set into the equivalence classes.
Partitions induced by equivalence (2.27) have a nice property that the partitions with smaller
indices kare the coarsenings of the partitions with the larger k’s. Moreover, these random parti-
tions are induced by a Markovian structure and can be seen as the states of the continuous-time
coalescent process at certain deterministic times introduced by Bolthausen & Sznitman (1998).
Indeed, for an arbitrary index set I, let E(I)be the set of all equivalence relations on I. The
sets E(I)are compact, if seen as the subsets of the product spaces {0,1}I×I(we equip the set
{0,1}with the discrete topology). Finally, let Γbe the Markov process
Γ:={Γ(t):Ω→E(
N
)}t∈
R
+
with the following properties.
2.2 Limiting objects 39
1. The initial state is Γ(0) = Π0, where Π0∈E(
N
)is the partition of
N
into singleton sets.
2. Coalescence property: for any s,t∈
R
+, whenever s<t, we have Γ(t)Γ(s), i.e., the
partition Γ(s)is not finer than Γ(t).
3. Finite-dimensional traces: for any Ib
N
, let ΓIbe the trace of the process Γon the set
E(I). Let aI:={aI(Π,Π0)}Π,Π0∈E(I)be the transition rate matrix (Q-matrix) of the Markov
process Γ(I)defined as follows. For Π,Π0∈E(I), denoting N:=cardI, define
aIΠ,Π0:=
(N−2)N−2
k−2−1,Π0is obtained by gluing k∈[2;N]∩
N
classes
of Πtogether;
−∑E6=E0aI(E,E0),Π=Π0;
0,otherwise.
Definition 2.2.3 (the Bolthausen-Sznitman coalescent).As is shown in (Bolthausen & Sznitman,
1998, Theorem 1.2), the above properties uniquely identify the distribution of the continuous-
time pure-jump Markov process Γwhich is then called the Bolthausen-Sznitman coalescent.
See Figure 2.1 for a realisation of the trace of the Bolthausen-Sznitman coalescent on a finite
set.
Fig. 2.1. A realisation of the trace of the Bolthausen-Sznitman coalescent on I:={1,2,3,4}
Remark 2.2.3. An explicit representation of the Markov semigroup of the Bolhausen-Sznitman
coalescent is certainly also available, see (Bolthausen & Sznitman, 1998, Proposition 1.4).
40 2 Often used tools and some existing results
Properties
For k∈[0;n]∩
N
, let the partition Πk∈E(
N
)be the partition generated by the Bolthausen-
Sznitman equivalence “∼
k”. The following theorem motivates Definition 2.2.3 by relating the
distributions of the limiting ultrametric overlap distribution function with the Bolthausen-
Sznitman coalescent.
Theorem 2.2.6 (limiting ultrametric overlap and the Bolthausen-Sznitman coalescent).
1. The law of the “limiting ultrametric overlaps” vector Πn−1,...,Π1coincides with the law
of the following Bolthausen-Sznitman vector
Γ(t1),Γ(t2),...,Γ(tn−1),
where, for k ∈[1;n−1]∩
N
tk:=logxn
xk∈
R
+.
2. The coalescent Γand the point process N(ξ(x1,...,xn)) are independent.
See (Bolthausen & Sznitman, 1998, Theorem 2.2) for a proof.
The limiting overlaps and certain random permutations
So far, in the present chapter, we were dealing with an important ingredient of the RSB picture,
namely, with the scaling limits of the (unnormalised) GREM exponentials at the level of point
processes. This lead us to the RPC. The second important ingredient of the Parisi RSB picture
are the overlaps (see Section 1.4.5).
Recall (0.2). Given x∈Q0
n, define the limiting GREM overlap q :A2→[0;1]as
q(α1,α2):=qqL(α1,α2).(2.28)
We shall need also the randomised limiting GREM overlap q :
N
2→[0;1]as
q(i,j):=qqL(i,j).
Note that the above definitions allow also for another representation of the reordered limiting
GREM overlap
q(i,j) = q(π(i),π(j)).
Finally, define the filtered limiting GREM process {Y(α,t):α∈A,t∈
R
+}as follows
Y(α;t):=
n
∑
k=0
1
[qk;1](t)Wk([α]k;t∧qk+1−qk),(2.29)
where ∆qk:=qk+1−qkand {{Wk([α]k;t)}t∈
R
+}k∈[0;n]∩
N
,α∈Ais the family of independent (for
different indices αand k) standard 1-D Wiener processes. Definition (2.29) readily implies
CovhY(α(1),t),Y(α(2),s)i=t∧s∧q(α(1),α(2)).
2.2 Limiting objects 41
Now, similarly to (2.16) with the help of the filtered limiting GREM we reweight the initial
RPC weights ξas
˜
ξ(α;t):=ξ(α)exp(f(t,Y(α,t))),(2.30)
where we assume, for simplicity, that f:[0;1]×
R
→
R
is such that, for any t∈[0;1],f(t,·)∈
C(2)(
R
), and any c>0 we have R
R
expf(t,x)−cx2dx<∞and, moreover, that
sup
x∈
R
|∂xf(t,x)|+|∂2
xx f(t,x)|<+∞.
Further, the point process defined in (2.30) can also be reordered in a decreasing way, i.e., there
exists the mapping ˜
πt:
N
→Asuch that, for all i∈
N
, the following holds
˜
ξ(˜
πt(i);t)>˜
ξ(˜
πt(i+1);t).(2.31)
In what follows, we shall use the short-hand notations ˜
ξ(i;t):=˜
ξ(˜
πt(i);t),˜
Y(i,t):=Y(˜
πt(i),t)
and ˜qt:={˜qt(i,j):=q(˜
πt(i),˜
πt(j))}i,j∈
N
.
Theorem 2.2.7 (some limiting GREM properties).Given a discrete order parameter x ∈Q0
n,
we have
1. Independence #1: the normalised RPC point process N(ξ)is independent from the corre-
sponding randomised limiting GREM overlaps q.
2. Independence #2: the reordered filtered limiting GREM ˜
Y is independent from the corre-
sponding reordered weights ˜
ξ.
3. The change of measure: given I b
N
, let µI(·|q)be the conditional distribution of {Y(i,1)}i∈I,
and ˜
µI(·|q)be the conditional distribution of {˜
Y(i,1)}i∈Iboth conditional on q. Then
d˜
µI(·|q)
dµI(·|q)=
n
∏
k=0
∏
i∈I/∼
kexpxkf(xk+1,Y(i,xk+1))−f(xk,Y(i,xk)),
where the innermost product is taken over all equivalence classes on the index set I induced
by the equivalence ∼
k.
4. The averaging property: the function f :[0;1]×
R
→
R
satisfies (1.36), if and only if
{ξ(α)exp(f(t,Y(α;t)))}α∈A,˜qt∼{ξ(α)exp(f(s,Y(α;s)))}α∈A,˜qs,
for all s,t∈[0;1].
See, e.g., Arguin (2007); Bolthausen & Sznitman (1998) for a proof.
Proposition 2.2.1 (expected mutial overlap distribution, Bolthausen & Sznitman (1998); Ruelle
(1987)).For any k ∈[1;n+1]∩
N
, we have
E
hN(ξ)⊗N(ξ)n(α(1),α(2))∈A2:qL(α1,α2)≤koi=xk.
See Bolthausen & Sznitman (1998); Ruelle (1987) for a proof.
42 2 Often used tools and some existing results
Some applications
We would like to point out that the Poisson-Dirichlet distribution and Bolthausen-Sznitman
coalescents appear in various applied contexts: in combinatorial stochastic structures, see, e.g.,
Arratia et al. (2003); Basdevant (2006); Bertoin (2006); Pitman (2006), in statistics, see, e.g.,
Teh et al. (2006). The structures are recently proved to emerge as equilibrium distributions for
certain systems of interacting diffusions, see, e.g., Chatterjee & Pal (2007).
2.3 Some results on Gaussian mean-field spin-glasses
2.3.1 The GREM
Recall the definition of the GREM from Section 1.4.3 and the decomposition (1.30) in particular.
Let ρ∈Q0
n. Assume that the process {GREM(σ):σ∈ΣN}has ρas its order parameter.
Assume that ρsatisfies (1.27). Define, for j,k∈[1;n+1]∩
N
,j<k, the “slopes” Aj,kof the
discrete order parameter xas
Aj,k:=qk−qj−1
2log2(xk−xj−1).
Define, further, the increasing sequence of indices {Jl}m
l=0⊂[0;n+1]∩
N
as follows. Start from
J0:=0, and define iteratively
Jl:=minnJ∈[Jl−1;n+1]∩
N
:AJl−1,J>AJ+1,k,for all k>Jo.
This subsequence of indices induces the following coarse-graining of the initial GREM. Define
the following (possibly) “coarse-grained” parameters
¯ql:=qJl−qJl−1,
¯xl:=xJl−xJl−1,
¯
γl:=s¯ql
(2log2)¯xl
.
Recall (2.12) and define the GREM scaling function uJ,N:
R
→
R
as
uJ,N(x):=
m
∑
l=1q(2log2)N¯xl¯ql−1
2√N¯
γllog(2log2)¯xl+x
N−1/2.
Define the rescaled GREM process as
GREM(σ):=u−1
J,N(GREM(σ)).(2.32)
Remark 2.3.1. Note that (2.32) is compatible with (2.24).
Define the point process of the rescaled GREM energies ENas
EN:=∑
σ∈ΣN
δGREM(σ).
2.3 Some results on Gaussian mean-field spin-glasses 43
The limiting object we are about to define now is a “logarithm” of the RPC point process. Given
K∈
R
+, assume that the point process P(1)(K)on
R
satisfies
P(1)(K)∼PPP(Kexp(−x)dx).
Consider the following collection of independent point processes
{P(k)
α1,...,αl−1(K)|α1,...,αl−1∈
N
;l∈[1;m]∩
N
}
such that
P(k)
α1,...,αk−1(K)∼P(1)(K).
Given K∈
R
m
+, define the limiting GREM point process Pm(K)on
R
mas follows
Pm(K):=∑
α∈Am
δ(P(1)(K1;α1),P(2)
α1(K2;α2),...,P(m)
α1,α2,...,αm−1(Km;αm)).
Now we are ready to state the following generalisation of Theorem 2.2.4.
Theorem 2.3.1 (the limit of the GREM point process).Assume that the “slopes” {γl}lform a
decreasing sequence, i.e., for all l ∈[1;m]∩
N
, we have
γl>γl+1.
Then there exists some K =K(x)∈(0;1]msuch that
EN−−→
N↑∞Z
R
mδγ1e1+...+γmemPm(K;de1,...,dem).
Moreover, Kl=1if and only if, for all x ∈(xl−1;xl), we have
[0;1]23(x,ρ(x)) /∈∂convΓ(ρ),
where Γ(ρ)is the sub-graph of the function ρ, i.e.,
Γ(ρ):=(x0,ρ(x)) ∈[0;1]2:x0≤x,x∈[0;1].
See (Bovier & Kurkova, 2004a, Theorem 1.5) for a proof.
Remark 2.3.2. Theorem 2.2.4 is a simple particular case of Theorem 2.3.1.
Theorem 2.3.1 allows for a complete characterisation of the limiting distribution of the
GREM’s partition function. To formulate the result, we need the β-dependent threshold l(β)∈
[0;m]∩
N
such that above this threshold (l>l(β)) all coarse grained levels lof the limiting
GREM are in a “frozen state” (“low temperature regime”). Below this threshold (l≤l(β)) the
levels are in the “high temperature regime”. Namely, define
l(β):=max{l∈[1;m]∩
N
:γlβ>1},
and let l(β):=0, if ¯
γ1β≤1.
44 2 Often used tools and some existing results
Theorem 2.3.2 (the limiting distribution of the GREM partition function ).Given β∈
R
+, there
exists the constant C(β;x)∈(0;1]such that one of the following two cases holds.
1. If l(β) = 0, then
ZN(β)
2Nexp(β2N/2)
w
−−−→
N↑+∞C(β,x).
2. If l(β)>0, then
exp l(β)
∑
l=1−βNq(2log2)¯ql¯xl+β¯
γllog((8log2)πN¯xl)
−N∑
k=Jl(β)+1β2qk/2+(2log2)¯xl
ZN(β)w
−−−→
N↑+∞
C(β,x)Z
R
l(β)exphβ¯
γ1x1+¯
γ2x2+...+¯
γl(β)xl(β)i
×Pl(β)(K(x);de1,...,del(β)).
Moreover, C(β;x) = 1, if and only if β¯
γl(β)+1<1.
See (Bovier & Kurkova, 2004a, Theorem 1.7) for a proof.
2.3.2 The SK model with multidimensional spins
Mean-field spin-glass models with multidimensional (Heisenberg) spins were considered in
the theoretical physics literature, see, e.g., Sherrington (2007) and references therein. Rigorous
results are, however, rather scarce. An early example is the result of Fr¨
ohlich & Zegarli´
nski
(1987), where the bounds on the free energy in the high temperature regime are obtained. Meth-
ods of stochastic analysis and large deviations were used by Toubol (1998) to identify the limit-
ing distribution of the partition function and also to obtain some information about the geometry
of the Gibbs measure for small β. More recent treatments of the high temperature regime using
very different methods are due to Talagrand (2000a), see also (Talagrand, 2003, Section 2.13).
The importance of the SK model with multidimensional spins for understanding the ultrametric-
ity of the original model of Sherrington & Kirkpatrick (1975) (which corresponds to d=1 and µ
being the Rademacher measure in the above notations) was emphasised by Talagrand (2007b).
The most advanced recent study of spin-glass models with multidimensional spins was at-
tempted by Panchenko & Talagrand (2007b), where the multidimensional spherical spin-glass
model was considered. The authors combined the techniques of Panchenko (2005b); Talagrand
(2006b) to obtain partial results on the ultrametricity and also get some information on the local
free energy for their model.
Multidimensional spins with compact support
Equip
R
dwith the Euclidean norm. Assume Σ:=B(0,√d)⊂
R
d. Given σ(1),σ(2)∈ΣN, define
the matrix R(σ(1),σ(2))∈
R
d×dwith the following entries
R(σ(1),σ(2))u,v:=1
N∑
i=1
σ(1)
i,uσ(2)
i,v.
2.3 Some results on Gaussian mean-field spin-glasses 45
Theorem 2.3.3 (existence of the RS order parameter).There exists L >0such that if Lβd≤1,
then there exist matrices Q(1),Q(2)∈
R
d×dsuch that
E
hGN(β)hkR(σ,σ)−Q(2)k2
2ii≤K(d)
N,
E
hGN(β)⊗GN(β)hkR(σ(1),σ(2))−Q(1)k2
2ii≤K(d)
N.
See (Talagrand, 2003, Theorem 2.13.1) for a proof.
The following theorem provides additional information on the matrices Q(1),Q(2). Assume
Q(2)is a symmetric non-negative definite matrix. There exists the Gaussian family {Yu}d
u=1with
covariance
E
[YuYv] = β2Q(2)
u,v.
Define the function E:
R
d→
R
as follows
E(x):=exp√2hx,Yi+β2h(Q(2)−Q(1))x,xi.
Theorem 2.3.4 (representation of the RS order parameters).Assume that βLd ≤1. We have
Q(1)
u,v=
E
1
Z2ZxuE(x)dµ(x)ZxvEdµ(x),(2.33)
Q(2)
u,v=
E
1
ZZxuxvE(x)dµ(x),(2.34)
where
Z:=ZE(x)dµ(x).
Moreover,
p(β) =
E
logZΣ
E(x)dµ(x)−β2
2kQ(2)k2
2−kQ(1)k2
2.
See (Talagrand, 2003, Theorems 2.13.2 and 2.13.3) for proofs.
One-dimensional spins with compact support
Let d=1. Given u∈
R
+, let us generalise slightly the objects of Section 1.5.3, namely let
Q0(u)⊂[0;1][0;u]be the set of all non-decreasing right-continuous piece-wise constant func-
tions x:[0;u]→[0;1]with x(0) = 0, x(u) = 1, such that they have only a finite number of
jumps. Let also Q0
n(u)⊂Q0(u)be the subset of order parameters with exactly n∈
N
jumps.
Assume the real sequence {cN}N∈
N
satisfies cN−−−→
N↑+∞0. Let {HN(σ)}σ∈ΣNbe the family of
Gaussian processes such that, for all σ(1),σ(2)∈ΣN, we have
1
N
E
hHN(σ(1))HN(σ(2))i−ξR(σ(1),σ(2))≤cN,(2.35)
46 2 Often used tools and some existing results
where the function ξ:
R
→
R
, for all x>0, satisfies the following (symmetry and convexity)
conditions
ξ(0) = 0,ξ(x) = ξ(−x),ξ00(x)>0.
A particular example of the Hamiltonian satisfying (2.35) is Derrida’s p-spin interaction Hamil-
tonian (cf. (1.15), p∈
N
)
HN,p(σ):=p−1/2N−(p−1)/2N
∑
i1,...,ip=1
gi1,...,ipσi1···σip,(2.36)
where {gi1,...,ip}N
i1,...,ip=1are i.i.d. standard normal random variables.
Let [d;D]⊂
R
be the smallest interval such that
µσ2∈[d;D]=1.
Assume {εN}∞
N=1⊂
R
+is such that εN−−−→
N↑+∞0. Define the set of all configurations with ap-
proximately the same self-overlap
UN(u):=σ∈ΣN:|R(σ,σ)−u|≤εN.
Define the corresponding local free energy
pN(u,εN):=1
N
E
logZN(u,εN),
where
ZN(u,εN):=ZUN(u)
expHN(σ)dµ⊗N(σ).
Given u∈[d;D], consider an arbitrary x∈Q0
n(u)and λ∈
R
. Consider, further, the family of
independent Gaussian random variables {zk}n
k=0such that
Varzk=ξ0(qk+1)−ξ0(qk).
Define the sequence of functionals {Xk:Q0
n(u)×
R
→
R
}n+1
k=0iteratively as follows. Start from
Xn+1(x,λ):=logZΣexp σ
n
∑
k=0
zn+λσ2!dµ(σ),
and continue, for k∈{n,n−1,...,0}, recursively
Xk(x,λ):=1
xk
log
E
expxkXk+1(x,λ).
Note that the functional X0(x,λ)is deterministic. Define the functional P:Q0
n(u)×
R
→
R
as
follows
2.3 Some results on Gaussian mean-field spin-glasses 47
P(x,λ):=−λu+X0(x,λ)−1
2
n
∑
k=1
xkθ(qk+1)−θ(qk),
where
θ(q):=qξ0(q)−ξ(q).(2.37)
Define the local Parisi free energy as
P(ξ,u):=inf
(x,λ)∈Q0(u)×
R
P(x,λ).
Finally, consider the (global) Parisi free energy
P(ξ):=sup
u∈[d;D]
P(ξ,u).
Theorem 2.3.5 (the saddle point Parisi formula).Given u ∈[d;D], there exists the vanishing
sequence {εN}N∈
N
such that
lim
N→+∞pN(u,εN) = P(ξ,u).
Moreover,
p(β) = P(ξ).
See (Panchenko, 2005b, Theorems 1 and 2) for proofs.
Multiple spherical spin-glass models
In this paragraph, deviating from the rest of the thesis, we shall consider the model with spins
σ∈ΣN, where ΣNis a non-product space (namely, a suitably chosen Euclidean sphere). Con-
sider the Gaussian Hamiltonian of the spherical spin-glass model1{HN(σ)}σ∈S(0,√N)satisfying
condition (2.35). Recall that (2.36) is a particular example of this situation. Define the free en-
ergy of the spherical spin-glass model as follows
pN(β,h):=1
N
E
logZS(0,√N)
expβHN(σ)+h
N
∑
i=1
σidλN(σ),(2.38)
where λN∈M1S(0,√N)is the uniform distribution on S(0,√N).
Consider x∈Q0
n, where Q0
nis defined in (1.26). The limiting free energy for (2.38) was
computed by Crisanti & Sommers (1992) and then the computation was made rigorous by
Talagrand (2006a) using the methods of Talagrand (2006b).
Given a real parameter b>1, define the sequences {dl}n
l=1and {Dl}n
l=1as
dk:=
n
∑
k=l
xkξ0(qk+1)−ξ0(qk),
Dl:=b−dl.
1Sometimes also referred to as “the spherical SK model”
48 2 Often used tools and some existing results
Define the Parisi functional P(β,h):Q0
n×(1;+∞)→
R
for the spherical spin-glass model as
P(β,h;x,b):=1
2 b−1−logb+1
D1
(h2+ξ0(q1))+
n
∑
k=1
1
xl
log Dl+1
Dl
−
n
∑
k=1
xlθ(ql+1)−θ(ql)!.
(See (2.37) for the definition of θ:[0;1]→
R
.) Also let us introduce the Crisanti-Sommers
functional C S (β,h):Q0
n→
R
as follows. Set δk:=∑n
l=kxl(ql+1−ql)and define
C S (β,h;x):=1
2 h2δ1+q1
δ1
+
n−1
∑
k=1
1
xk
log δk
δk+1
+logδn+
n
∑
k=1
xk(ξ(qk+1)−ξ(qk))!.
The following result is obtained by Talagrand (2006a).
Theorem 2.3.6 (the Parisi formula for the spherical spin-glass model).We have
p(β,h) = P(β,h):=inf
x∈Q0,b>1
P(β,h;x,b)
=inf
x∈Q0
C S (β,h;x).
See Talagrand (2006a) for a proof.
Now, we are ready to consider multiple copies of the spherical model defined above. Let
Q∈Sym(d)be the non-negative definite matrix with elements Qu,v∈[−1;1]and Qu,u=1, for
all u,v∈[1;d]∩
N
. Given ε>0, consider the following set of configurations having almost the
same self-overlap given by Q, namely
U(ε):=nσ∈S(0,√N)d:|R(σ,σ)u,v−Qu,v|≤ε,for all u,v∈[1;d]∩
N
o.
Consider β1,...,βd>0 and h1,...,hd∈
R
. Denote β:={βu}d
u=1and h:={hu}d
u=1. Define the
local free energy on U(ε)of the dcopies of the spherical model as follows
p(d)
N(β,h,U(ε)) :=1
N
E
"ZU(ε)
exp d
∑
u=1
βuHN(σ·,u)+
d
∑
u=1hu
N
∑
u=1
σi,u!#.
It is easy to see that
p(d)
N(β,h,U(ε)) ≤
d
∑
u=1
pN(βu,hu)
which implies that
lim
N→∞p(d)
N(β,h,U(ε)) ≤
d
∑
u=1
P(βu,hu).
Consider the following inequality
2.3 Some results on Gaussian mean-field spin-glasses 49
lim
ε→+0lim
N→∞p(d)
N(β,h,U(ε)) <
d
∑
u=1
P(βu,hu).(2.39)
An interesting result of Panchenko & Talagrand (2007b) indicates that, in general, inequality
(2.39) does not hold, as one can infer from the following theorem.
Theorem 2.3.7 (local Parisi formula for the multidimensional spherical model (Panchenko &
Talagrand, 2007b)).Assume we are in the situation of the Hamiltonian (2.36) with p =2, i.e.,
the classical SK case. Suppose βu:=β>1, for all u ∈[1;d]∩
N
, and minru≥1, where {ru}d
u=1
are the eigenvalues of the self-overlap matrix Q.
Then
lim
ε→+0lim
N→+∞p(d)
N(β,0,U(ε)) = d·P(β,0).
See (Panchenko & Talagrand, 2007b, Theorem 2) for a proof.
2.3.3 The Aizenman-Sims-Starr comparison scheme for the SK model
The AS2scheme (Aizenman et al., 2003, 2007) gives an intrinsic way to obtain variational
upper bounds on the free energy in the SK model. The scheme is also based on a comparison
between two Gaussian processes. The first process is the sum of the original SK Hamiltonian
Xand a GREM-inspired process indexed by additional index space A:=
N
n. The second one
is another GREM-inspired process indexed by the extended configuration space ΣN×A. The
scheme uses a comparison functional defined on Gaussian processes indexed by the extended
configuration space equipped with the product measure between the original a priori measure
and Ruelle’s probability cascade. The role of the comparison functional in the AS2scheme is
played by a free energy functional acting on the Gaussian processes indexed by the extended
configuration space.
It is interesting to note that Panchenko & Talagrand (2007a) have reexpressed Guerra’s
scheme for the SK model using the RPC.
Increments of the free energy of the SK model
Suppose HN(σ)is the SK Hamiltonian given by (1.19) and PN(β,h)is given by (1.4).
Theorem 2.3.8 (superadditivity of the free energy).For any N,M∈
N
,
PN(β,h)+PM(β,h)≤PN+M(β,h).(2.40)
See Guerra & Toninelli (2002) for a proof.
Theorem 2.3.8 immediately gives (by Theorem 2.1.2) the existence of the thermodynamic
limit (cf. Question 0.0.1). The next theorem builds upon more subtle consequences of superad-
ditivity, namely on relation (2.10).
Theorem 2.3.9 (incremental representation of the free energy).We have
lim
N→∞lim
M→∞
1
N
E
logZN+M(β,h)
ZN(β,h)=p(β,h).
See, e.g., (Aizenman et al., 2007, Corollary 3.5) for a proof.
50 2 Often used tools and some existing results
Remark 2.3.3. Note that the free energy pN(β,h)is sharply concentrated around its expec-
tation – a fact that was first established by a martingale argument by Pastur & Shcherbina
(1991) and then made more precise by the general machinery of concentration of measure, see,
e.g., (Talagrand, 2003, Corollary 2.2.5). Any of these two facts together with Theorem 2.3.8 is
sufficient to conclude that
pN(β,h)−−−→
N↑+∞p(β,h),almost surely and in L1.
Let {C(α)}α∈ΣM,{B(α)}α∈ΣM, and {A(α,σ)}σ∈ΣN,α∈ΣMbe three independent Gaussian
processes with the following correlation structures2
CovhC(α(1)),C(α(2))i=M
N+MRM(α(1),α(2))2
,
CovhA(σ(1),α(1)),A(σ(2),α(2))i=2N
N+MRN(σ(1),σ(2))RM(α(1),α(2)),
CovhB(α(1)),B(α(2))i=N
N+MRM(α(1),α(2))2
.
As a short but crucial computation shows, Theorem 2.3.9 implies the following result.
Theorem 2.3.10 (second incremental representaion of the free energy).We have
lim
N→∞lim
M→∞
1
N
E
log
∑(σ,α)∈ΣN×ΣMη(α)expβ√MA(σ,α)+h∑N
i=1σi
∑α∈ΣNη(α)expβ√MB(α)
=p(β,h),(2.41)
where
η(α):=expβ√MC(α)+h
M
∑
i=1
σi.(2.42)
See, e.g., (Bovier, 2006, Section 11.3.1) or (Aizenman et al., 2007, Theorem 4.1) for a proof.
Theorem 2.3.10 suggests that in (2.41) instead of (2.42) one can consider arbitrary random
weights η:={η(α)}α∈A, where Ais a countable set and ∑α∈Aη(α)<∞almost surely.
Assume also that the Gaussian processes A:={A(σ,α)}σ∈ΣN
α∈A
and B:={B(α)}α∈Ahave the
following correlation structures
CovhA(σ(1),α(1)),A(σ(2),α(2))i=2RN(σ(1),σ(2))q(α(1),α(2)),
CovhB(α(1)),B(α(2))i=q(α(1),α(2))2,(2.43)
where q:={q(α(1),α(2))}α(1),α(2)∈Ais the non-negative definite kernel with q(α,α) = 1, for
all α∈A. Assume, further, that the following functional
GN(β,h,η,q):=1
N
E
"∑(σ,α)∈ΣN×Aη(α)expβ√NA(σ,α)+h∑N
i=1σi
∑α∈Aη(α)expβ√NB(α)#
is well defined.
2It is easy prove the existence of such processes.
2.3 Some results on Gaussian mean-field spin-glasses 51
Theorem 2.3.11 (the AS2variational bound).For any N ∈
N
and any η, q as above, we have
pN(β,h)≤GN(β,h,η,q).
Moreover,
GN(β,h,η,q)−pN(β,h)
=β2
2Z1
0
G(β,h,t,η,q)⊗G(β,h,t,η,q)q(α(1),α(2))−R(σ(1),σ(2))2dt,
where G(β,h,t,η,q)∈M1ΣN×Ais the Gibbs measure induced by the Hamiltonian H(t):
Σ×A→
R
defined as follows
H(t;σ,α):=√t(HN(σ)+B(α))+√1−tA(σ,α)+h
N
∑
i=1
σ,
where t ∈[0;1]. More precisely, for a measurable f :ΣN×A→
R
, we have
G(β,h,t,η,q)[ f] = 1
Z∑
(σ,α)∈Σ×A
f(σ,α)η(α)expβ√NH(t;σ,α),
where Z is the usual probabilistic normalisation factor.
See (Aizenman et al., 2007, Theorem 4.1 ) for a proof.
Comparison with the GREM, relation with the Parisi functional
Consider the following comparison functional
Φ(η)[T]:=
E
log ∑
(σ,α)∈ΣN×A
η(α)expβ√NT (σ,α)+h
N
∑
i=1
σ
,(2.44)
where T:={T(σ,α)}σ∈ΣN
α∈A
is an arbitrary Gaussian process such that (2.44) exists.
We then obviously have
GN(β,h,η,q) = Φ(η)[A]−Φ(η)[B].
Given x∈Q0
n, let η:=ξ(x1,...,xn)be the corresponding RPC. Let, further, q=q(x)be the
limiting GREM overlap generated by x(cf. (2.28)). Let A=A(q)and B=B(q)be the corre-
sponding Gaussian processes with covariances given by (2.43).
Theorem 2.3.12 (comaprison with the GREM, relation with the Parisi functional).Let the func-
tion fx(q,y):[0;1]×
R
→
R
be the solution of (1.36).
Then we have
Φ(η)[A] = fx(0,0),
Φ(η)[B] = β2
2Z1
0
qx(q)dq,
and consequently
GN(β,h,η,q) = P(β;x).
See, e.g., (Aizenman et al., 2007, Lemma 6.2) for a proof.
Note that Theorems 2.3.12 and 2.3.11 immediately imply the upper bound in (1.37), i.e.,
p(β,h)≤inf
x∈Q0
P(β,x).(2.45)
3
The Aizenman-Sims-Starr scheme for the SK model with
multidimensional spins
In this chapter, we are mainly concerned with the question of the validity of the Parisi formula
in the case where spins take values in a d-dimensional Riemannian manifold. We address the
issue of extending the approach of Aizenman, Sims and Starr to the multidimensional spins.
3.1 Introduction
The recent rigorous proof of the celebrated Parisi formula for the free energy of the SK model,
due to Talagrand (2006b), based on the ingenious interpolation schemes of Guerra (2003) and
Aizenman et al. (2003) constitutes one of the major recent achievements of probability theory.
Recently, these results have been generalised to spherical SK-models (Talagrand, 2006a) and to
models with spins taking values in a bounded subset of
R
(Panchenko, 2005b).
A particular case (d=1, µwith bounded support) of the model we are considering here was
treated by Panchenko (2005b). He used the techniques of Talagrand (2006b) to prove that in the
case d=1 upper and lower bounds on the free energy coincide (cf. (3.13) and (3.20) in this
chapter). However, the results of (Panchenko, 2005b, Section 5 and the proofs of Theorems 2,
5 and 9) are based on relatively detailed differential properties of the optimal Lagrange multi-
pliers in the saddle point optimisation problem of interest. These properties are harder to obtain
in multidimensional situations such as that we are dealing with here. In fact, as we show in
Theorems 3.1.1 and 3.1.2, one can obtain the same saddle point variational principles without
invoking the detailed properties of the optimal Lagrange multipliers. This is achieved using a
quenched LDP of the G¨
artner-Ellis type.
Definition of the model
We refer to the introduction of this thesis for the definition of the SK model with multidimen-
sional spins. (See, in particular, (0.8) and (0.9).)
Throughout the chapter we assume that we are given a large enough probability space
(Ω,F,
P
)such that all random variables under consideration are defined on it. Without further
notice we shall assume that all Gaussian random variables (vectors and processes) are centred.
We shall be interested mainly in the free energy
pN(β):=1
NlogZΣN
expβ√NX(σ)dµ⊗N(σ),(3.1)
where β≥0 is the inverse temperature and µ∈Mf(Σ)is some arbitrary (not necessarily uni-
form or discrete) finite a priori measure. We assume that the a priori measure µis such that
54 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
(3.1) is finite. We shall be interested in proving bounds on the thermodynamic limits of these
quantities, e.g., on
p(β):=lim
N↑+∞pN(β).(3.2)
Remark 3.1.1. Note that there is no need to include the additional external field terms into the
Hamiltonian (0.8), since they could be absorbed into the a priori measure µ.
Main results
Below we introduce the notations, assumptions and formulate the main results of this chapter.
Assumption 3.1.1. Suppose that the configuration space Σis bounded and such that 0∈
intconvΣ, where convΣdenotes the convex hull of Σ.
The examples listed below verify this assumption:
1. Multicomponent Ising spins. Σ={−1;1}d– the discrete hypercube.
2. Heisenberg spins. Σ=σ∈
R
d:kσk2=1– the unit Euclidean sphere.
3. Σ=σ∈
R
d:kσk2≤1– the unit Euclidean ball.
Remark 3.1.2. The boundedness assumption can be relaxed and replaced by concentration
properties of the a priori measure. In Section 5.2 we will exemplify this in the case of a Gaussian
a priori distribution. In general a subgaussian distribution will suffice.
Consider the space of all symmetric matrices Sym(d):=Λ∈
R
d×d|Λ=Λ∗. Denote
Sym+(d):={Λ∈Sym(d)|Λ0},
where the notation Λ0 means that the matrix Λis non-negative definite. We equip the space
Sym(d)with the Frobenius (Hilbert-Schmidt) norm
kMk2
F:=
d
∑
u,v=1
M2
u,v,M∈Sym(d).
We shall also denote the corresponding (tracial) scalar product by h·,·i. For r>max{kσk2
2:
σ∈Σ}, define
U:=U∈Sym(d)|U0,kUk2≤r.
We will call the set Uthe space of the admissible self-overlaps. In analogy to the usual overlap
in the standard SK model, we define, for two configurations, σ(i)= (σ(i)
1,σ(i)
2,...,σ(i)
N)∈ΣN,
i=1,2, the (mutual) overlap matrix RN(σ(1),σ(2))∈
R
d×dwhose entries are given by
RN(σ(1),σ(2))u,v:=1
N
N
∑
i=1
σ(1)
i,uσ(2)
i,v,u,v∈[1;d]∩
N
.(3.3)
Fix an overlap matrix U ∈U. Given a subset V⊂U, define the set of the local configurations,
3.1 Introduction 55
ΣN(V):=σ∈ΣN|RN(σ,σ)∈V.
Next, define the local free energy
pN(V):=1
NlogZΣN(V)
eβ√NX(σ)dµ⊗N(σ).(3.4)
We also define
p(V):=p(β,V):=lim
N↑+∞pN(V),(3.5)
where the existence of the limit follows from a result of (Guerra & Toninelli, 2003, Theorem 1).
Consider a sequence of matrices Q:={Q(k)∈Sym(d)}n+1
k=0such that
0=:Q(0)≺Q(1)≺...≺Q(n+1):=U,(3.6)
where the ordering is understood in the sense of the corresponding quadratic forms. Consider
in addition a partition of the unit interval x:={xk}n+1
k=0, i.e.,
0=:x0<x1< ... < xn+1:=1.(3.7)
Let {z(k)}n
k=0be a sequence of independent Gaussian d-dimensional vectors with
Covhz(k)i=Q(k+1)−Q(k).
Given Λ∈Sym(d), define
Xn+1(x,Q,U,Λ):=logZΣexp√2βn
∑
k=0
zk,σ+hΛσ,σidµ(σ).(3.8)
Define, for k∈{n,...,0}, by a descending recursion,
Xk(x,Q,U,Λ):=1
xk
log
E
z(k)expxkXk+1(x,Q,U,Λ) (3.9)
with
X0(x,Q,U,Λ):=
E
z(0)X1(x,Q,U,Λ),(3.10)
where
E
z(k)[·]denotes the expectation with respect to the σ-algebra generated by the random
vector z(k).
Remark 3.1.3. Section 4.1.4 contains the more general framework of dealing with the recursive
quantities (3.10) which in particular brings to light the links with certain non-linear parabolic
PDEs. For these PDEs the recursion (3.1.2) is closely related to an iterative application of the
well-known Hopf-Cole transformation, see, e.g., Evans (1998).
Define the local Parisi functional as
f(x,Q,U,Λ):=−hΛ,Ui−β2
2
n
∑
k=1
xkkQ(k+1)k2
F−kQ(k)k2
F+X0(x,Q,U,Λ).(3.11)
56 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
Assumption 3.1.2 (Hadamard squares).We shall say that a sequence, {Q(i)}n
i=1, of matrices
satisfies Assumption 3.1.2, if
Q(1)2≺...≺Q(n)2≺Q(n+1)2.(3.12)
Remark 3.1.4. The above assumption on the matrix order parameters Qis necessary only
to employ the AS2scheme. In contrast, Guerra’s scheme (Theorems 4.1.1 and 4.16) does not
require the above assumption.
One may verify that the matrices qand ρin (Talagrand, 2003, Theorems 2.13.1 and 2.13.2)
correspond to the matrices Q(1)and Q(2)of this chapter (n=1). (See also (3.23) below.) Further-
more, a straightforward application of the Cauchy-Schwarz inequality shows that the matrices
qand ρactually satisfy Assumption 3.1.2. We also note that in the simultaneous diagonalisa-
tion scenario in which the matrices in (3.6) are diagonalisable in the same orthogonal basis (see
Sections 4.2.3 and 5.2.2) this assumption is also satisfied.
The first main result of the present chapter uses the AS2scheme to establish the upper bound
on the limiting free energy p(β)in terms of the saddle point problem for the local Parisi func-
tional (3.11).
Theorem 3.1.1. For any closed set V⊂Sym(d), we have
p(V)≤sup
U∈V∩U
inf
(x,Q,Λ)f(x,Q,Λ,U),(3.13)
where the infimum runs over all x satisfying (3.7), all Qsatisfying both (3.6) and Assump-
tion 3.1.2, and all Λ∈Sym(d).
We were not able to prove in general that the r.h.s. of (3.13) gives also the lower bound to
the thermodynamic free energy. See, however, Theorem 5.1.1 for a positive example.
To formulate the lower bound on (3.2) we need some additional definitions.
Let the comparison index space be A:=
N
n. Given α(1),α(2)∈A, define
Q(α(1),α(2)):=Q(qL(α(1),α(2))),(3.14)
where qL(α(1),α(2))is defined in (0.2) Given a d×d-matrix Mand p∈
R
, we denote by Mp
the d×d-matrix with entries
Mpu,v:= (Mu,v)p.
The matrix valued lexicographic overlap (3.14) can be used to construct the multidimensional
(d≥1) versions of the GREM (see, e.g., Bovier & Kurkova (2007) and references therein
for a review of the results on the one-dimensional case of the model). Here we shall need the
following two GREM-inspired real-valued Gaussian processes: A:={A(σ,α)}σ∈ΣN,α∈Aand
B:={B(α)}α∈Awith covariance structures
E
hA(σ(1),α(1))A(σ(2),α(2))i=2hR(σ(1),σ(2)),Q(α(1),α(2))i,
E
hB(α(1))B(α(2))i=kQ(α(1),α(2))k2
F.
3.1 Introduction 57
Note that the process Acan be represented in the following form:
A(σ,α) = 2
N1/2N
∑
i=1hAi(α),σii,(3.15)
where {Ai:={Ai(α)}α∈A}N
i=1are the i.i.d. (for different indices i) Gaussian
R
d-valued pro-
cesses with the following covariance structure: for i∈[1;N]∩
N
, for all α(1),α(2)∈Aand all
u,v∈[1;d]∩
N
assume that the following holds
E
hAi(α(1))uAi(α(2))vi=Q(α(1),α(2))u,v.
Given t∈[0;1], we define the interpolating AS2Hamiltonian
Ht(σ,α):=√t(X(σ)+B(α))+√1−tA(σ,α).(3.16)
Next, we define the random probability measure πN∈M1(ΣN×A)through
πN:=µ⊗N⊗ξ,
where ξ=ξ(x)is the RPC. We denote by {ξ(α)}α∈Athe enumeration of the atom locations
of the RPC and consider the enumeration as a random measure on A(independent of all other
random variables around). Define the local AS2Gibbs measure GN(t,x,Q,U,V)by
GN(t,x,Q,U,V)[ f]:=1
ZN(t,V)ZΣN(V)×A
f(σ,α)e√NβHt(σ,α)dπN(σ,α),(3.17)
where f:ΣN×A→
R
is an arbitrary measurable function for which the right-hand side of
(3.17) is finite. For V⊂U, define the AS2remainder term as
RN(x,Q,U,V)
:=−1
2Z1
0
E
hGN(t,x,Q,U,V)⊗GN(t,x,Q,U,V)hkRN(σ(1),σ(2))−Q(α(1),α(2))k2
Fiidt.
(3.18)
We define also the limiting AS2remainder term
R(x,Q,U):=lim
ε↓+0lim
N↑∞
RN(x,Q,B(U,ε)) ≤0,(3.19)
where B(U,ε)is the ball with centre Uand radius ε. (The existence of the limiting remainder
term is proved in Theorem 3.1.2.)
The second main result of this chapter uses the AS2scheme to establish a lower bound
on (3.2) in terms of the same saddle point Parisi-type functional as in the upper bound which
includes, however, the non-positive remainder term (3.19). In one-dimensional situations Tala-
grand (2006b) and Panchenko (2005b), respectively, have shown that the corresponding error
term vanishes on the optimiser of the Parisi functional.
58 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
Theorem 3.1.2. For any open set V⊂Sym(d), we have
p(V)≥sup
U∈V∩U
inf
(x,Q,Λ)[f(x,Q,Λ,U)+R(x,Q,U)],(3.20)
where the infimum runs over all x satisfying (3.7), all Λ∈Sym(d), and all Qsatisfying both
(3.6) and Assumption 3.1.2.
Remark 3.1.5. The comparison scheme of Guerra (2003) (see also more recent accounts Tala-
grand (2007a), Guerra (2005) and Aizenman et al. (2007)) is also applicable to our model and
is covered by our quenched LDP approach, see Theorems 4.1.1 and 4.16 for the formal state-
ments. Guerra’s scheme seems to be more amenable (compared to the Aizenman-Sims-Starr
one) for Talagrand’s remainder estimates (Talagrand, 2006b), see Section 4.3. The scheme is
based on the following interpolation
e
Ht(σ,α):=√tX(σ)+ √1−tA(σ,α)(3.21)
which induces the corresponding local Gibbs measure (3.17) and remainder term (3.18) by
substituting (3.16) with (3.21). Guerra’s scheme does not include the process B and, hence,
does not require Assumption 3.1.2. Recovering the terms corresponding to ΦN(x,U)[B](see,
(3.105)) in the Parisi functional requires then a short additional calculation (Lemma 4.1.1).
Note that the results of (Talagrand, 2003, Theorems 2.13.2 and 2.13.3) imply that at least in
the high temperature region (i.e., for small enough β) the Parisi formula for the SK model with
multidimensional spins is valid with n=1
p(β) = f(x,Q∗,0,U∗) = sup
U∈U
inf
(Q,Λ)f(x,Q,Λ,U),(3.22)
where the matrices Q∗(2)=U∗and Q∗(1)solve the following system of equations:
∂Q(2)
u,v
f(x,Q∗,0,U∗) = 0,u,v∈[1;d]∩
N
,
∂Q(1)
u,v
f(x,Q∗,0,U∗) = 0,u,v∈[1;d]∩
N
.(3.23)
Note that the system (3.23) coincides with the mean-field equations obtained in Theorem 2.3.4.
3.2 Some preliminary results
3.2.1 Covariance structure
Our definition of the overlap matrix in (3.3) is motivated by the fact that, as can be seen from a
straightforward computation
E
hXN(σ(1))XN(σ(2))i=
d
∑
u,v=1RN(σ(1),σ(2))u,v2=kRN(σ(1),σ(2))k2
2,(3.24)
that is, the the covariance structure of the process XN(σ)is given by the square of the Frobenius
(Hilbert-Schmidt) norm of the matrix RN(σ(1),σ(2)). The basic properties of the overlap matrix
are summarised in the following proposition.
3.2 Some preliminary results 59
Proposition 3.2.1. We have, for all σ(1),σ(2),σ∈ΣN,
1. Matrix representation. RN(σ(1),σ(2)) = 1
Nσ(1)∗σ(2).
2. Symmetry #1. Ru,v
N(σ(1),σ(2)) = Rv,u
N(σ(2),σ(1)).
3. Symmetry #2. Ru,v
N(σ,σ) = Rv,u
N(σ,σ).
4. Non-negative definiteness #1. RN(σ,σ)0.
5. Non-negative definiteness #2.
RN(σ(1),σ(1))RN(σ(1),σ(2))
RN(σ(1),σ(2))∗RN(σ(2),σ(2))0.
6. Suppose U :=RN(σ(1),σ(1)) = RN(σ(2),σ(2)), then
kR(σ(1),σ(2))k2
F≤kUk2
F.
Proof. The proof is straightforward. ut
3.2.2 Concentration of measure
The following concentration of measure result for the free energy is standard.
Proposition 3.2.2. Let (Σ,S)be a Polish space. Suppose µis a random finite measure on Σ.
Suppose, moreover, that X(σ),σ∈Σis the family of Gaussian random variables independent
of µwhich possesses a bounded covariance, i.e.,
there exists K >0such that sup
σ(1),σ(2)∈Σ|Cov(X(σ(1)),X(σ(2)))|≤K.(3.25)
Assume that
f(X):=logZΣeX(σ)dµ(σ)<∞.
Then
P
{|f(X)−
E
[f(X)]|≥t}≤2exp−t2
4K.
Remark 3.2.1. An analogous result was given in a somewhat more specialised case in Panchenko
(2005b).
Proof. This is an adaptation of the proof of (Talagrand, 2003, Theorem 2.2.4). We can not apply
the comparison Theorem 3.2.5 directly, so we resort to the basic interpolation argument as stated
in Proposition 2.1.1. For j=1,2, let the processes Xj(·)be the two independent copies of the
process X(·). For t∈[0;1], let
Xj,t:=√tXj+√1−tX
and
Fj(t):=logZΩexpXj,t(σ)dµ(σ).
60 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
For s∈
R
, let
ϕs(t):=
E
exps(F1−F2).
Hence, differentiation gives
˙
ϕs(t) = s
E
exps(F1−F2)(˙
F1−˙
F2)(3.26)
(the dots indicate the derivatives with respect to t) and also
˙
Fj(t) =1
2ZΣexpXj,t(σ)dµ(σ)−1
×ZΣt−1/2Xj(σ)−(1−t)−1/2X(σ)expXj,t(σ)dµ(σ).(3.27)
Now, we substitute (3.27) back to (3.26) and apply Corollary 2.1.1 to the result. After some
tedious but elementary calculations we get
˙
ϕs(t) =s2
E
"exps(F1−F2)ZΣexpX1,t(σ)dµ(σ)ZΣexpX2,t(σ)dµ(σ)−1
ZΣCov(X(σ(1)),X(σ(2)))expX1,t(σ(1))+ X2,t(σ(2))dµ(σ(1))dµ(σ(2)).
Thus, thanks to (3.25), we obtain
˙
ϕs(t)≤Ks2ϕs(t).
The conclusion of the theorem follows now exactly as in the proof of (Talagrand, 2003, Theo-
rem 2.2.4).
ut
We now apply this general result to the our model and also to the free energy-like functional
of the GREM-inspired process A.
Proposition 3.2.3. Suppose Σ⊂B(0,r), for r >0. For Ω⊂ΣN, denote
PSK
N(β,Ω):=logZΩexp√NβXN(σ)dµ⊗N(σ),
and
PGREM
N(β,Ω):=logZΩ×A
expβ√2
N
∑
i=1hAi(α),σiidπN(σ,α).
Then, for all Ω⊂ΣN, we have
1. For any t >0,
P
PSK
N(β,Ω)−
E
PSK
N(β,Ω)>t≤2exp−t2
4β2r4N.(3.28)
3.2 Some preliminary results 61
2. For any t >0,
P
PGREM
N(β,Ω)−
E
PGREM
N(β,Ω)>t≤2exp−t2
8β2r4N.(3.29)
Proof. 1. We would like to use Proposition 3.2.2. By (3.24) and the Cauchy-Bouniakovsky-
Schwarz inequality, we have, for all N∈
N
,σ(1),σ(2)∈ΣN, that
Cov(XN(σ(1),σ(2))) = kRN(σ(1),σ(2))k2
F=1
N2
N
∑
i,j=1hσ(1)
i,σ(1)
jihσ(2)
i,σ(2)
ji≤r4.(3.30)
Hence, for all N∈
N
and all subsets Ωof ΣN, we obtain
sup
σ(1),σ(2)∈Σ|Cov(X(σ(1)),X(σ(2)))|≤r4.
Thus (3.28) is proved.
2. We fix an arbitrary N∈
N
,σ(1),σ(2)∈ΣN,α(1),α(2)∈A. We have
Cov(A(σ(1),α(1)),A(σ(2),α(2))) =
E
hA(σ(1),α(1))A(σ(2),α(2))i
=
N
∑
i=1hQ(α(1),α(2))σ(1)
i,σ(2)
ii.
Bound (3.30) implies that, for any U∈U, we have kUk2≤r2. Since Q(α(1),α(2))∈U,
we obtain
|hQ(α(1),α(2))σ(1)
i,σ(2)
ii|≤kQ(α(1),α(2))k2kσ(1)
ik2kσ(2)
ik2
≤kQ(α(1),α(2))k2r2≤r4.
Therefore, using Proposition 3.2.2, we obtain (3.29).
ut
3.2.3 Gaussian comparison inequalities for free energy-like functionals
We begin by recalling well-known integration by parts formula which is the source of many
comparison results for functionals of Gaussian processes.
The following proposition connects the computation of the derivative of the free energy with
respect to the parameter that linearly occurs in the Hamiltonian with a certain Gibbs average for
a replicated system.
Proposition 3.2.4. Consider a Polish measure space (Σ,S)and a random measure µon it. Let
X={X(σ)}σ∈Σand Y :={Y(σ)}σ∈Σbe two independent Gaussian real-valued processes. For
u∈
R
, we define
Hu(σ):=uX(σ)+Y(σ).
Assume that, for all u ∈[a,b]b
R
, we have
62 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
Zexp(Hu(σ))dµ(σ)<∞,ZX(σ)exp(Hu(σ))dµ(σ)<∞
almost surely, and also that
E
logZexp(Hu(σ))dµ(σ)<∞.
Then we have
d
du
E
logZeHu(σ)dµ(σ)=u
E
[G(u)⊗G(u)[VarX(σ)−
E
[X(σ),X(τ)]]],
where G(u)is the random element of M1(Σ)which, for any measurable f :Σ→
R
, satisfies
G(u)[ f] = 1
Z(u)Zf(σ)exp(Hu(σ))dµ(σ).
Proof. We write
d
dulogZeHu(σ)dµ(σ) = ZX(σ)eHu(σ)
Zu(β)dµ(σ),(3.31)
where Zu(β):=ReβHu(σ)dµ(σ). The main ingredient of the proof is the Gaussian integration
by parts formula. Denote, for τ∈Σ,e(τ):=
E
[X(σ)Hu(τ)]. By (2.3), we have
E
"X(σ)eHu(σ)
Zu(β)#=
E
"∂X eHu(σ)
ReHu(τ)dµ(τ)!(X;e)#.(3.32)
Due to the independence, we have
E
[X(σ)Hu(τ)] = u
E
[X(σ),X(τ)].
Henceforth, the computation of the directional derivative in (3.32) amounts to
∂
∂t"eHu(σ)+tuVar(σ)
ReHu(τ)+tuCov(σ,τ)dµ(τ)#
=ZeHu(σ)dµ(σ)−2uVarX(σ)eHu(σ)ZeHu(τ)dµ(τ)
−eHu(σ)ZuCov[X(σ),X(τ)]eHu(τ)dµ(τ).(3.33)
Substituting the r.h.s. of (3.33) into (3.31), we obtain the assertion of the proposition. ut
The following proposition gives a short differentiation formula, which is useful in getting
comparison results between the (free energy-like) functionals of Gaussian processes.
3.2 Some preliminary results 63
Proposition 3.2.5. Let (X(σ))σ∈Σ,(Y(σ))σ∈Σbe two independent Gaussian processes as be-
fore. Set
Ht(σ):=√tX(σ)+ √1−tY(σ).
Assume that
ZeHt(σ)dµ(σ)<∞,ZX(σ)eHt(σ)dµ(σ)<∞,
ZY(σ)eHt(σ)dµ(σ)<∞
almost surely, and also that, for all t ∈[0;1],
E
logZeHt(σ)dµ(σ)<∞.
Then we have
E
logZeX(σ)dµ(σ)=
E
logZeY(σ)dµ(σ)
−1
2Z1
0
G(t)⊗G(t)hVarX(σ(1))−VarY(σ(1))
−CovhX(σ(1)),X(σ(2))i−CovhY(σ(1)),Y(σ(2))iidt,(3.34)
where G(t)is the random element of M1(Σ)which, for all measurable f :Σ→
R
, satisfies
G(t)[ f] = 1
Z(t)ZΣ
f(σ)exp(Ht(σ))dµ(σ).(3.35)
Proof. Let us introduce the process
Wu,v(σ):=uX(σ)+vY(σ).
Hence,
Ht(σ) = W√t,√1−t(σ).(3.36)
Thus
d
dt
E
logZeHt(σ)dµ(σ)=1
21
√t
∂
∂u
E
logZeWu,v(σ)dµ(σ)
−1
√1−t
∂
∂v
E
logZeWu,v(σ)dµ(σ)u=√t,v=√1−t
.
Applying Proposition 3.2.4 and R1
0·dtto the previous formula, we conclude the proof. ut
64 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
3.3 Quenched G¨
artner-Ellis type LDP
In this section, we derive a quenched LDP under measure concentration assumptions. Theo-
rems 3.3.1 and 3.3.2 give the corresponding LDP upper and lower bounds, respectively. The
proofs of the LDP bounds will be adapted to get the proofs of the upper and lower bounds
on the free energy of the SK model with multidimensional spins. However, they may be of
independent interest.
Note that the existing “level-2” quenched large deviation results of Comets (1989) are ap-
plicable only to a certain class of mean-field random Hamiltonians which are required to be
“macroscopic” functionals of the joint empirical distribution of the random variables represent-
ing the disorder and the independent spin variables. The SK Hamiltonian can not be repre-
sented in such form, since the interaction matrix consists of i.i.d. random variables. Moreover,
it is assumed in Comets (1989) that the Hamiltonian has the form HN(σ) = NV(σ), where
{V(σ)}σ∈ΣNis a random process taking values in some fixed bounded subset of
R
. Since the
Hamiltonian of our model is a Gaussian process, this assumption is also not satisfied, due to the
unboundedness of the Gaussian distribution.
3.3.1 Quenched LDP upper bound
Throughout this section we impose the following.
Assumption 3.3.1. Suppose {QN}∞
N=1is a sequence of random measures on a Polish space
(X,X). Assume that there exists some L >0such that for any QN-measurable set A ⊂Xwe
have
P
logQN(A)−
E
logQN(A)>t≤exp−t2
LN .
Note that this assumption will hold in the cases we are interested in due to Proposition 3.2.2.
Lemma 3.3.1. Suppose {QN}∞
N=1is a sequence of random measures on a Polish space (X,X)
and {Ar⊂X:r∈{1,...,p}}is a sequence of QN-measurable sets such that, for some absolute
constant L >0, we have
P
logQN(Ar)−
E
logQN(Ar)>t≤exp−t2
LN .(3.37)
Then we have
lim
N↑+∞
1
N
E
logQNp
[
r=1
Ar−max
r∈{1,...,p}
E
logQN(Ar)=0.(3.38)
Proof. First, (3.37) gives
P
max
r∈{1,...,p}logQN(Ar)−
E
logQN(Ar)≥t≤2pexp−t2
LN .
Since, for a,b∈
R
p, the following elementary inequality holds
max
rar−max
rbr≤max
r|ar−br|,
3.3 Quenched G¨
artner-Ellis type LDP 65
we get
P
max
r∈{1,...,p}logQN(Ar)−max
r∈{1,...,p}
E
logQN(Ar)≥t≤2pexp−t2
LN .
The last equation in turn implies that
1
N
E
max
r∈{1,...,p}logQN(Ar)−max
r∈{1,...,p}
E
logQN(Ar)≤2pZ+∞
0
exp−Nt2
Ldt,(3.39)
and the r.h.s. of the previous formula vanishes as N↑∞.
ut
Let QN∈M(X),N∈
N
be a family of random measures on (X,X). Define the Laplace
transform
LN(Λ):=ZX
eNhx,ΛidQN(x).
Suppose that, for all Λ∈
R
d, we have
I(Λ):=lim
N↑∞
1
N
E
logLN(Λ)∈
R
=
R
∪{−∞,+∞}.(3.40)
Define the Legendre transform
I∗(x):=inf
Λ[−hx,Λi+I(Λ)].(3.41)
Define, for δ>0,
I∗
δ(x):=maxI∗(x)+δ,−1
δ.(3.42)
Lemma 3.3.2. Suppose
0∈intD(I):=int{Λ:I(Λ)<+∞}.(3.43)
Then
1. The mapping I∗(·):X→
R
is upper semi-continuous and concave.
2. For all M >0,
{x∈X:I∗(x)≤M}is a compact.
Proof. 1. Since, for all Λ∈D(I), the linear mappings
x7→−hΛ,xi+I(Λ)
are obviously concave, the infimum of this family is upper semi-continuous and concave.
2. See, e.g., den Hollander (2000) for the proof.
ut
66 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
Theorem 3.3.1. Suppose that
1. The family {QN}satisfies condition (3.38).
2. Condition (3.40) is satisfied.
3. Condition (3.43) is satisfied.
Then, for any closed set V⊂
R
d, we have
lim
N↑∞
1
N
E
logQN(V)≤sup
x∈V
I∗(x).(3.44)
Proof. 1. Suppose at first that Vis a compact.
Thanks to (3.41), for any x∈X, there exists Λ(x)∈Xsuch that
−hx,Λ(x)i+I(Λ(x)) ≤I∗
δ(x).(3.45)
For any x∈X, there exists a neighbourhood A(x)⊂Xof xsuch that
sup
y∈A(x)hy−x,Λ(x)i≤δ.
By compactness, the covering Sx∈YA(x)⊃Vhas the finite subcovering, say Sp
r=1A(xr)⊃
V. Hence,
1
NlogQN(V)≤1
Nlog p
[
r=1
QN(A(xr))!.(3.46)
Applying condition (3.38), we get
lim
N↑∞
1
N
E
max
r∈{1,...,p}logQN(A(xr))−max
r∈{1,...,p}
E
1
NlogQN(A(xr))≤0.(3.47)
By the Chebyshev inequality,
QN(A(x)) ≤QN{y∈X:hy−x,Λ(x)i≤δ}
≤e−δNZX
eNhy−x,Λ(x)idQN(y)
=e−δNe−Nhx,Λ(x)iLN(Λ(x)).(3.48)
Hence, (3.48) together with (3.45) yields
lim
N↑+∞
1
N
E
logQN(A(xr))≤lim
N↑+∞−hxr,Λ(xr)i+1
NlogLN(Λ(xr))−δ
=−hxr,Λ(xr)i+I(Λ(xr))−δ
≤I∗
δ(xr)−δ.(3.49)
Combining (3.46), (3.47), (3.49), we obtain
lim
N↑+∞
1
N
E
logQN(V)≤max
r∈{1,...,p}I∗
δ(xr)−δ
≤sup
x∈V
I∗
δ(x)−δ.
Taking δ↓+0 limit, we get the assertion of the theorem.
3.3 Quenched G¨
artner-Ellis type LDP 67
2. Let us allow now the set Vto be unbounded. We first prove that the family QNis quenched
exponentially tight. For that purpose, let
RN(M):=1
N
E
hlogQN(X\[−M;M]d)i,
and denote
R(M):=lim
N↑+∞RN(M).
We want to prove that
lim
M↑+∞R(M) = −∞.(3.50)
Fix some u∈{1,...,d}. Suppose δu,p∈{0,1}is the standard Kronecker symbol. Let eu∈
R
dbe an element of the standard basis of
R
d, i.e., for all p∈{1,...,d}, we have
(eu)p:=δu,p.
Thanks to the Chebyshev inequality, we have
QN{xu≤−M}≤e−NM Z
R
de−Nhx,euidQN(x),a.s. (3.51)
Now, we get
Z
R
de−Nhx,euidQN(x) = 1
LN(Λe)Z
R
deNhx,Λe−euidQN(x)
=LN(Λe−eu)
LN(Λe),a.s. (3.52)
Hence, combining (3.51) and (3.52), we obtain
1
N
E
logQN{xu≤−M}≤−M+IN(Λe−eu)−IN(Λe).(3.53)
Using the same argument, we also get
1
N
E
logQN{xu≥M}≤−M+IN(Λe+eu)−IN(Λe).(3.54)
We obviously have
RN(M)≤1
N
E
hlogQNd
[
u=1
({xu≤−M}∪{xu≥M})i.(3.55)
Applying condition (3.38) to (3.55), we get
lim
N↑+∞
1
N
E
hlogQNd
[
u=1{xu≤−M}∪{xu≥M}
68 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
−max
u∈{1,...,d}maxn
E
[logQN({xu≤−M})],
E
[logQN({xu≥M})]oi≤0.(3.56)
Applying (3.53) and (3.54) in (3.56), we get
lim
N↑+∞
1
N
E
hlogQNd
[
u=1
({xu≤−M}∪{xu≥M})i
≤−M−I(Λe)+ max
u∈{1,...,d}max{I(Λe−eu),I(Λe+eu)}.(3.57)
The bound (3.57) assures (3.50). Now, since we have (with the help of (3.38) and (3.44))
lim
N↑+∞
1
N
E
logQN(V)≤lim
N↑+∞
1
N
E
hlogQN((V∩[−M;M]d)∪(X\[−M;M]d))i
≤maxsup
x∈(V∩[−M;M]d)
I∗(x),R(M),(3.58)
the assertion of the theorem follows from (3.50) by taking the limM↑+∞in the bound (3.58).
ut
3.3.2 Quenched LDP lower bound
Suppose that, for some Λ∈
R
dand all N∈
N
, we have
ZX
eNhy,ΛidQN(y)<+∞.
Let e
QN,Λ∈M(X)be the random measure defined by
e
QN,Λ(A) = ZA
eNhy,ΛidQN(y),(3.59)
for any QNmeasurable A⊂X.
Lemma 3.3.3. Suppose the family of random measures QNsatisfies the following assumptions.
1. Measure concentration. There exists some L >0such that, for any QN-measurable set A ⊂
X, we have
P
logQN(A)−
E
logQN(A)>t≤exp−t2
LN .
2. Tails decay condition. Let
C(M):={x∈X:kxk<M}.
There exists p ∈
N
such that
lim
K↑+∞lim
N↑∞Z+∞
0
P
1
Nlog e
QN,Λ(X\C(Np)) >−K+tdt=0.
3.3 Quenched G¨
artner-Ellis type LDP 69
3. Non-degeneracy. The family of the sets nBj⊂X:j∈{1,...,q}osatisfies the following
condition
there exists some j0∈{1,...,q}such that lim
N↑∞
1
N
E
hlog e
QN,Λ(Bj0)i>−∞.(3.60)
Then, for any Λ∈
R
d, we have
lim
N↑∞
1
N
E
log e
QN,Λq
[
j=1
Bj−max
j∈{1,...,q}
E
hlog e
QN,Λ(Bj)i≤0.(3.61)
Remark 3.3.1. The polynomial growth choice of M =MN:=Npmade in assumption (2) of
the above lemma corresponds certainly to the concrete form of the exponential concentration
in assumption (1). Both assumptions together with the proof can be adapted to the weaker
concentration bounds.
Proof. We fix some j∈ {1,...,q}. Take an arbitrary ε>0, M>0 and denote JM,ε:=
Z
∩
[−kΛkM/ε;kΛkM/ε]. Consider, for i∈JM,ε, the following closed sets
Ai,j:={x∈Bj:(j−1)ε≤hΛ,xi≤ jε}.
We get
1
Nlog e
QN,Λq
[
j=1
Bj≤1
Nlog e
QN,Λq
[
j=1
Bj∩C(M)∪(X\C(M))
≤1
Nmaxnmax
j∈{1,...,q}log e
QN,Λ(Bj∩C(M)),
log e
QN,Λ(X\C(M))o+log(q+1)
N.(3.62)
We have
1
Nlog e
QN,Λ(Bj∩C(M)) ≤1
Nlog ∑
i∈JM,ε
eNiεQN(Ai,j)
≤max
i∈{1,...,p}iε+1
NlogQN(Ai,j)+log(cardJM,ε)
N.(3.63)
Denote
αN(ε):=max
j∈{1,...,q}max
i∈JM,εiε+1
NlogQN(Ai,j),
and
βN:=max
j∈{1,...,q}
E
hlog e
QN,Λ(Bj)i,
e
βN(ε):=max
j∈{1,...,q}
E
"max
i∈JM,εiε+1
NlogQN(Ai,j)#,
70 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
γN(M):=1
Nlog e
QN,Λ(X\C(M)).
We also have
1
Nlog e
QN,Λ(Bj)≥1
Nlog e
QN,Λ(Bj∩C(M))
≥max
i∈JM,ε(i−1)ε+1
NlogQN(Ai,j)
=max
i∈JM,εiε+1
NlogQN(Ai,j)−ε.(3.64)
Due to condition (1), we have
P
nαN(ε)−e
βN(ε)>to≤cardJM,εqexp−Nt2
L.(3.65)
We put M:=MN:=Np, and we get
cardJM,ε≤2kΛkM/ε+1
≤2kΛkNp/ε+1.(3.66)
Let
XN(M,ε):=max{γN(M),αN(ε)}−βN,
then we have
P
{XN(K,ε)>t}≤
P
{γN(M)>βN+t}+
P
{αN(ε)>βN+t}.(3.67)
Due to property (3.60), there exists K>0 such that we have
P
{γN(M)>βN+t}≤
P
{γN(M)>−K+t}.(3.68)
Thanks to (3.64), we have
P
{αN(ε)>βN+t}≤
P
{αN(ε)>e
βN(ε)+t−ε}.(3.69)
For t>ε, we apply (3.65) and (3.66) to (3.69) to obtain
P
{αN(ε)>βN+t}≤(2kΛkNp/ε+1)qexp−Nt2
L.(3.70)
Combining (3.62) and (3.63), we get
E
log e
QN,Λq
[
j=1
Bj−max
j∈{1,...,q}
E
hlog e
QN,Λ(Bj)i
≤
E
XN(M,ε)+log(q+1)
N+log(2kΛkNp/ε+1)
N.(3.71)
3.3 Quenched G¨
artner-Ellis type LDP 71
Now, (3.67), (3.68) and (3.70) imply
E
XN(M,ε)≤Z+∞
0
P
XN(M,ε)>tdt
≤Z+∞
ε
P
XN(M,ε)>tdt+ε
≤Z+∞
ε
P
{γN(M)>−K+t}dt
+(2kΛkNp/ε+1)qZ+∞
εexp−Nt2
Ldt+ε.(3.72)
Therefore, taking sequentially limN↑+∞, limK↑+∞and limε↑+0in (3.72), we arrive at
lim
N↑∞
E
XN(M,ε)≤0.(3.73)
The bound (3.73) together with (3.71) implies the assertion of the lemma.
ut
Let ˆ
QN,Λbe the (random) probability measure defined by
ˆ
QN,Λ:=e
QN
LN(Λ).
Lemma 3.3.4. Suppose that the measure QNsatisfies the assumptions of the previous lemma.
Then (3.61) is valid also for ˆ
QN,Λ.
Proof. Similar to the one of the previous lemma. ut
Remark 3.3.2. Recall that a point x ∈Xis called an exposed point of the concave mapping I∗
if there exists Λ∈
R
dsuch that, for all y ∈X\{x}, we have
I∗(y)−I∗(x)<hy−x,Λi.(3.74)
Theorem 3.3.2. Suppose
1. The family QN:N∈
N
⊂M(
R
d)satisfies the assumptions of Lemma 3.3.3.
2. G⊂Xis an open set.
3. /0 6=E(I∗)⊂D(I∗)is the set of the exposed points of the mapping I∗.
4. Condition (3.43) is satisfied.
Then
lim
N↑+∞
1
N
E
logQN(G∩E)≥sup
x∈G
I∗(x).(3.75)
Proof. Let B(x,ε)be a ball of radius ε>0 around some arbitrary x∈X. It suffices to prove
that
lim
ε↓+0lim
N↑∞
1
N
E
logQN(B(x,ε))≥I∗(x).(3.76)
72 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
Indeed, since we have
QN(G)≥QN(B(x,ε)),(3.77)
applying 1
Nlog(·), taking the expectation, taking limN↑+∞,ε↓+0 and taking the supremum over
x∈Gin (3.77), we get (3.75).
Take any x∈G∩E. Then we can find the corresponding vector Λe=Λe(x)∈
R
dorthog-
onal to the exposing hyperplane at the point x, as in (3.74). Define the new (“tilted”) random
probability measure ˆ
QNon
R
dby demanding that
dˆ
QN
dQN
(y) = 1
LN(Λe)eNhy,Λei.(3.78)
Moreover, we have
1
N
E
logQN(B(x,ε))=1
N
E
logZB(x,ε)
dQN(y)
=1
N
E
logLN(Λe)+1
N
E
ZB(x,ε)
e−Nhy,Λeidˆ
QN(y)
≥1
N
E
logLN(Λe)−hx,Λei−εkΛek2+1
N
E
log ˆ
QN(B(x,ε)).
Hence,
lim
ε↓+0lim
N↑∞
1
N
E
logQN(B(x,ε))≥[−hx,Λei+I(Λe)]+ lim
ε↓+0lim
N↑∞
1
N
E
log ˆ
QN(B(x,ε)).
Since we have
−hx,Λei+I(Λe)≥I∗(x),
in order to show (3.76) it remains to prove that
lim
ε↓+0lim
N↑∞
1
N
E
log ˆ
QN(B(x,ε))=0.(3.79)
The Laplace transform of ˆ
QNis
ˆ
LN(Λ) = LN(Λ+Λe)
LN(Λe).
Hence, we arrive at
ˆ
I(Λ) = I(Λ+Λe)−I(Λe).
Moreover, we have
ˆ
I∗(x) = I∗(x)+hx,Λei−I(Λe).(3.80)
3.3 Quenched G¨
artner-Ellis type LDP 73
By the assumptions of the theorem, the family QNsatisfies the assumptions of Lemma 3.3.3.
Hence, due to Lemma 3.3.4, the family ˆ
QNsatisfies (3.38). Thus we can apply Theorem 3.3.1
to obtain
lim
N↑+∞
1
N
E
hlog ˆ
QN(
R
d\B(U,ε))i≤sup
y∈U\B(x,ε)
ˆ
I∗(y).(3.81)
Lemma 3.3.2 implies that there exists some x0∈X\B(x,ε)(note that x06=x) such that
sup
y∈X\B(x,ε)
ˆ
I∗(y) = ˆ
I∗(x0).
Since Λeis an exposing hyperplane, using (3.80), we get
ˆ
I∗(x0) = I∗(x0)+hx0,Λei−I(Λe)
≤[I∗(x0)+hx0,Λei]−[I∗(x)+hx,Λei]<0,(3.82)
and hence, combining (3.81) and (3.82), we get
lim
N↑+∞
1
N
E
hlog ˆ
QN(
R
d\B(x,ε))i<0.
Therefore, due to the concentration of measure, we have almost surely
lim
N↑+∞
1
Nlog ˆ
QN(
R
d\B(x,ε)) <0
which implies that, for all ε>0, we have almost surely
lim
N↑+∞
ˆ
QN(
R
d\B(x,ε)) = 0,
and (3.79) follows by yet another application of the concentration of measure.
ut
Corollary 3.3.1. Suppose that in addition to the assumptions of previous Theorem 3.3.2 we
have
1. I(·)is differentiable on intD(I).
2. Either D(I) = Xor
lim
Λ→∂D(I)k∇I(Λ)k= +∞.
Then E(I∗) =
R
d, consequently
lim
N↑+∞
1
N
E
logQN(G)≥sup
x∈G
I∗(x).
Proof. The proof is the same as in the classical G¨
artner-Ellis theorem (see, e.g., den Hollander
(2000)).
ut
74 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
3.4 The Aizenman-Sims-Starr comparison scheme
In this section, we shall extend the AS2scheme to the case of the SK model with multidi-
mensional spins and prove the Theorems 3.1.1 and 3.1.2, as stated in Section 3.1. We use the
Gaussian comparison results of Section 3.2.3 in the spirit of AS2scheme in order to relate
the free energy of the SK model with multidimensional spins with the free energy of a certain
GREM-inspired model. Comparing to Aizenman et al. (2003), due to more intricate nature of
spin configuration space, some new effects occur. In particular, the remainder term of the Gaus-
sian comparison non-trivially depends on the variances and covariances of the Hamiltonians
under comparison. To deal with this obstacle, we use the idea of localisation to the configura-
tions having a given overlap (cf. (3.4)). This idea is formalised by adapting the proofs of the
quenched G¨
artner-Ellis type LDP obtained in Section 3.3.
3.4.1 Naive comparison scheme
We start by recalling the basic principles of the AS2comparison scheme (see, e.g., (Bovier,
2006, Chapter 11)). It is a simple idea to get the comparison inequalities by adding some ad-
ditional structure into the model. However, the way the additional structure is attached to the
model might be suggested by the model itself. Later on we shall encounter a real-world use of
this trick. Let (Σ,S)and (A,A)be Polish spaces equipped with measures µand ξ, respec-
tively. Furthermore, let
X:={X(σ)}σ∈Σ,A:={A(σ,α)}σ∈Σ,
α∈A
,B:={B(σ)}α∈A
be independent real-valued Gaussian processes. Define the comparison functional
Φ[C]:=
E
logZΣ×A
eC(σ,α)d(µ⊗ξ)(σ,α),(3.83)
where C:={C(σ,α)}σ∈Σ
α∈A
is a suitable real-valued Gaussian process. Theorem 4.1 of Aizen-
man et al. (2007) is easily understood as an example of the following observation. Suppose
Φ[X]is somehow hard to compute directly, but Φ[A]and Φ[B]are manageable. We always
have the following additivity property
Φ[X+B] = Φ[X]+Φ[B].(3.84)
Assume now that
Φ[X+B]≤Φ[A](3.85)
which we can obtain, e.g., from Proposition 3.2.5. Combining (3.84) and (3.85), we get the
bound
Φ[X]≤Φ[A]−Φ[B].(3.86)
3.4.2 Free energy upper bound
Let V⊂Sym(d)be an arbitrary Borell set.
3.4 The Aizenman-Sims-Starr comparison scheme 75
Remark 3.4.1. Note that Uis closed and convex.
Let
ΣN(V):=σ∈ΣN:RN(σ,σ)∈V
=σ∈ΣN:RN(σ,σ)∈V∩U.(3.87)
Let us define the local comparison functional ΦN(x,V)as follows (cf. (3.83))
ΦN(x,V)[C]:=1
N
E
hlogπNh
1
ΣN(V)expβ√NCii,(3.88)
where C:={C(σ,α)}σ∈Σ
α∈A
is a suitable Gaussian process. Let us consider the following family
(N∈
N
) of random measures on the Borell subsets of Sym(d)generated by the SK Hamiltonian,
PN(V):=ZΣN(V)
eβ√NXN(σ)dµ⊗N(σ),
and consider also the following family of the random measures generated by the Hamiltonian
A(σ,α)
e
PN(V):=e
Px,Q,U
N(V):=ZΣN(V)×A
expβ√N
N
∑
i=1hAi(α),σiidπN(σ,α),(3.89)
where the parameters Qand Uare taken from the definition of the process A(α)(cf. (3.6)).
The vector xdefines the random measure ξ∈M(A)(cf. (3.7)), and, hence, also the measure
πN∈M(Σ×A).
Remark 3.4.2. To lighten the notation, most of the time we shall not indicate explicitly the
dependence of the following quantities on the parameters x, Q, U.
Consider (if it exists) the Laplace transform of the measure (3.89)
e
LN(Λ):=ZU
eNhU,Λide
PN(U).(3.90)
Let (if it exists)
e
I(Λ):=lim
N↑∞
1
N
E
hloge
LN(Λ)i.(3.91)
Define the following Legendre transform
e
I∗(U):=inf
x∈Q0(1,1),
Q∈Q0(U,d),
Λ∈Sym(d)
h−hU,Λi−ΦN(x,V)[B]+ e
I(Λ)i.(3.92)
Denote, for δ>0,
e
I∗
δ(U):=maxe
I∗(U)+δ,−1
δ.
Let
p(V):=lim
N↑+∞
1
N
E
logPN(V).(3.93)
76 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
Remark 3.4.3. Note that the result of Guerra & Toninelli (2003) assures the existence of the
limit in the previous formula.
Lemma 3.4.1. We have
1. The Laplace transform (3.90) exists. Moreover, for any Λ∈Sym(d), we have
ZV
eNhU,ΛidPN(U)
=ZΣN(V)
expNhΛ,RN(σ,σ)i+β√NX(σ)dµ⊗N(σ),(3.94)
ZV
eNhU,Λide
PN(U)
=ZΣN(V)×A
expNhΛ,RN(σ,σ)i+β√N
N
∑
i=1hAi(α),σiidπN(σ,α).(3.95)
2. The quenched cumulant generating function (3.91) exists in the N ↑∞limit, for any Λ∈
Sym(d). Moreover, for all N ∈
N
, we have
IN(Λ):=1
N
E
logLN(Λ)=X0(x,Q,Λ,U),(3.96)
that is IN(·)in fact does not depend on N.
Proof. 1. We prove (3.95), the proof of (3.94) is similar. Since Uis a compact, it follows that,
for arbitrary ε>0, there exists the following ε-partition of U
N(ε) = {Vr⊂U:r∈{1,...,K}}
such that SrVr=U,Vr∩Vs=/0, diamVr≤εand pick some Vr∈intVr, for all r6=s.
We denote
e
LN(Λ,ε):=
K
∑
r=1
eNhΛ,VriZΣN(Vr)×A
expβ√N
N
∑
i=1hAi(α),σiidπN(σ,α).
For small enough ε, we have
(1−2NkΛkε)eNhΛ,RN(σ,σ)i≤eNhΛ,Ui≤eNhΛ,RN(σ,σ)i(1+2NkΛkε).
Therefore, if we denote
b
LN(V,Λ):=ZΣN(V)×A
expNhΛ,RN(σ,σ)i+β√N
N
∑
i=1hAi(α),σiidπN(σ,α),
we get
(1−2NkΛkε)
K
∑
r=1b
LN(Vr,Λ)≤e
LN(Λ,ε)≤(1+2NkΛkε)
K
∑
r=1b
LN(Vr,Λ).
Hence,
(1−2NkΛkε)b
LN(U,Λ)≤e
LN(Λ,ε)≤(1+2NkΛkε)b
LN(U,Λ).(3.97)
Let ε↓+0 in (3.97) and we arrive at
e
LN(Λ) = b
LN(U,Λ).
That is, the existence of LN(Λ)and the representation (3.95) are proved.
3.4 The Aizenman-Sims-Starr comparison scheme 77
2. For all N∈
N
, we have, by the RPC averaging property (see, e.g., (Aizenman et al., 2007,
Theorem 5.4) or Theorem 4.1.3, property (4) below), that
1
N
E
hloge
LN(U,Λ)i=ΦN(x,U)A+NhΛ,RN(σ,σ)i=X0(x,Q,Λ,U).
ut
Proof of Theorem 3.1.1. In essence, the proof follows almost literally the proof of Theorem
3.3.1. The notable difference is that we apply the Gaussian comparison inequality (Proposi-
tion 3.2.5) in order to “compute” the rate function in a somewhat more explicit way.
Due to (3.87), we can without loss of generality suppose that Vis compact. For any δ>
0 and U∈V, by (3.92), there exists Λ(U,δ)∈Sym(d),x(U,δ)∈Q0(1,1)and Q(U,δ)∈
Q0(U,d)such that
−hU,Λ(U)i+e
I(Λ(U)) ≤e
I∗
δ(U).(3.98)
For any U∈V, there exists an open neighbourhood V(U)⊂Sym(d)of Usuch that
sup
V∈V(U)hV−U,Λ(U)i≤δ.
Fix some ε>0. Without loss of generality, we can suppose that all the neighbourhoods satisfy
additionally the condition diamV(U)≤ε. By compactness, the covering SU∈VV(U)⊃V
has a finite subcovering, say Sp
r=1V(U(r))⊃V. We denote the corresponding to this covering
approximants in (3.98) by {x(r)∈Q0(1,1)}p
r=1and {Q(r)∈Q0(U(r),d)}p
r=1. We have
1
NlogPN(V)≤1
Nlogp
[
r=1
PN(V(U(r))).(3.99)
Due to the concentration of measure Proposition 3.2.3, we can apply Lemma 3.3.1 and get
lim
N↑+∞
1
N
E
logPNp
[
r=1
V(U(r))−max
r∈{1,...,p}
E
hlogPN(V(U(r)))i=0.(3.100)
In fact, since we know that (3.93) exists, (3.100) implies that
lim
N↑+∞
1
N
E
hlogPNp
[
r=1
V(U(r))i=max
r∈{1,...,p}lim
N↑+∞
1
N
E
hlogPN(V(U(r)))i.(3.101)
For U(r),x=x(r),Q=Q(r), Proposition 3.2.5 gives
1
N
E
hlogPN(V(U(r)))i=1
N
E
hlog e
PN(V(U(r)))i−ΦN(x,U)[B]
+RN(x(r),Q(r),U(r),V(U(r)))+O(ε)
≤1
N
E
hlog e
PN(V(U(r)))i−ΦN(x,U)[B]+Kε,(3.102)
where K>0 is an absolute constant.
By the Chebyshev inequality and Lemma 3.4.1, we have
78 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
e
PN(V(U)) ≤e
PN{V∈U:hV−U,Λ(U)i≤δ}
≤e−δNZU
eNhV−U,Λ(U)ide
PN(V)
=e−δNe−NhU,Λ(U)ie
LN(Λ(U)).
Thus, using (3.102) and (3.98), we get
lim
N↑+∞
1
N
E
hlogPN(V(U(r)))i≤lim
N↑+∞−hU(r),Λ(U(r))i−Φ[B]+ 1
Nloge
LN(Λ(U(r)))−δ+Kε
=−hUr,Λ(Ur)i−Φ[B] + e
I(Λ(Ur))−δ+Kε
≤e
I∗
δ(Ur)−δ+Kε.(3.103)
Combining (3.99), (3.47), (3.103), we obtain
p(V) = lim
N↑+∞
1
N
E
logPN(V)≤max
r∈{1,...,p}lim
N↑+∞
1
N
E
hlogPN(V(U(r)))i
≤max
r∈{1,...,p}e
I∗
δ(U(r))+Kε−δ
≤sup
U∈Ve
I∗
δ(V)+Kε−δ.
Taking δ↓+0 and ε↓+0 limits, we get
p(V)≤sup
V∈Ve
I∗(U).(3.104)
The averaging property of the RPC (see, e.g., (Aizenman et al., 2007, Theorem 5.4) or property
(4) of Theorem 4.1.3) gives
ΦN(x,U)[B] = β2
2
n
∑
k=1
xkkQ(k+1)k2
F−kQ(k)k2
F.(3.105)
To finish the proof it remains to show that, for any fixed Λ∈Sym(d), we have
e
I(Λ) = X0(x,Q,Λ,U)
which is assured by Lemma 3.4.1. ut
3.4.3 Free energy lower bound
In this subsection, we return to the notations of Section 3.4.2.
Lemma 3.4.2. For any B⊂Sym(d)such that intB∩intU6=/0 there exists ∆⊂Σwith int∆6=
/0 such that
lim
N↑∞
1
N
E
ZΣN(B)×A
expNhΛ,RN(σ,σ)i+
N
∑
i=1hAi(α),σiidπN(σ,α)
≥logZ∆exph(β2U+Λ)σ,σidµ(σ)>−∞.(3.106)
3.4 The Aizenman-Sims-Starr comparison scheme 79
Proof. In view of (3.9), iterative application of the Jensen inequality with respect to
E
z(k)leads
to the following
E
Xn+1(x,Q,Λ,U)≤X0(x,Q,Λ,U).
Performing the Gaussian integration, we get
E
Xn+1(x,Q,Λ,U)≥logZ∆exph(β2U+Λ)σ,σidµ(σ),
where ∆⊂Σis such that µ(∆)>0 and {R(σ,σ):σ∈∆N}⊂B.ut
Define the following Legendre transform
b
I∗(U):=inf
x∈Q0(1,1),
Q∈Q0(U,d),
Λ∈Sym(d)
h−hU,Λi−Φ[B]+ e
I(Λ)+R(x,Q,U)i.(3.107)
Proof of Theorem 3.1.2. As it is the case with the proof of Theorem 3.1.1, this proof also fol-
lows in essence almost literally the proof of Theorem 3.3.2. The notable difference is that we
apply the Gaussian comparison in order to “compute” the rate function in a somewhat more
explicit way.
In notations of Theorem 3.3.2, we are in the following situation: X:=Sym(d)and Xis the
topology induced by any norm on Sym(d).
Let B(U,ε)be the ball (in the Hilbert-Schmidt norm) of radius ε>0 around some arbitrary
U∈V. Let us prove at first that
lim
ε↓+0lim
N↑∞
1
N
E
logPN(B(U,ε))≥b
I∗(U).(3.108)
Similarly to (3.102), for any (x,Q), we have
E
1
NlogPN(B(U,ε))
=1
N
E
hlog e
PN(B(U,ε))i−Φ[B]+RN(x,Q,U,B(U,ε))+O(ε).(3.109)
The random measure e
PNsatisfies the assumptions of Corollary 3.3.1. Indeed:
1. Due to representation (3.96), mapping I(·)is differentiable with respect to Λ. Henceforth
assumption (1) of the corollary is also fulfilled.
2. Let us note at first that, thanks to Proposition 3.2.3, we have D(I) =
R
d. Thus, the assump-
tion (2) of Corollary 3.3.1 is satisfied, as is condition (3.43).
Moreover, the assumptions of Lemma 3.3.3 are satisfied:
1. The concentration of measure condition is satisfied due to Proposition 3.2.3.
2. The tail decay is obvious since the family {e
PN:N∈
N
}has compact support. Namely, for
all N∈
N
, we have supp e
PN=U. Thus the measure e
QN,Λ(cf. (3.59)) generated by e
PNhas
the same support. Thus, supp e
QN,Λ=U.
80 3 The Aizenman-Sims-Starr scheme for the SK model with multidimensional spins
3. The non-degeneracy is assured by Lemma 3.4.2.
Hence, due to (3.109), arguing in the same way as in Theorem 3.3.2, we arrive at (3.108). Note
that the N↑+∞limit of RN(x,Q,U,B(U,ε)) exists, since in (3.109) the limits of the other two
N-dependent quantities exist due to Guerra & Toninelli (2003). The subsequent ε↓+0 limit of
the remainder term exists due to the monotonicity.
Finally, taking the supremum over U∈Vin (3.108), we get (3.20). ut
4
Estimates of Guerra’s remainder term in the SK model with
multidimensional spins
In this chapter, we extend the comparison scheme of Guerra (2003) to the case of the SK model
with multidimensional spins. We provide a short derivation of the remainder term which is
a by-product of this scheme (Theorem 4.1.4). We use the well-known properties of the RPC
and Bolthausen-Sznitman coalescent. This gives the remainder term a clear meaning in terms
of the averages with respect to the disorder induced by a change of measure. The change of
measure is induced by the reweighting of the RPC by means of the GREM-inspired process (cf.
Theorem 2.2.7).
We study the properties of the multidimensional Parisi functional by establishing a link
between the (multidimensional) Parisi functional and a certain class of non-linear PDEs, see
Propositions 4.2.1, 4.2.2 and Theorem 4.2.2. We extend the Parisi functional to a continuous
functional on a compact space (Theorems 4.2.1, 4.2.2). We show that the class of PDEs cor-
responds to the Hamilton-Jacobi-Bellman equations induced by a linear problem of diffusion
control (Proposition 4.2.4). Motivated by a problem posed by Talagrand (2006c), we show the
strict convexity of the local Parisi functional in some cases (Theorem 4.2.4).
We partially extend Talagrand’s methodology of estimating the remainder term to the multi-
dimensional setting (Proposition 4.3.1, Theorem 4.3.1).
4.1 Guerra’s comparison scheme
For the SK model, Guerra’s scheme gave historically the first way to obtain the variational
upper bound on the free energy in terms of the Parisi functional. The scheme is based on the
comparison between two Gaussian processes: the first one being the original SK Hamiltonian
(0.8) and the other one being a carefully chosen GREM inspired process indexed by σ∈ΣN. The
second important ingredient is a recursively defined non-linear comparison functional acting on
the Gaussian processes indexed by σ∈ΣN.
Talagrand (2006b) using Guerra’s scheme and the wealth of other ingenious analytical in-
sights showed that the variational upper bound is also the lower bound for the free energy in the
SK model. This established, hence, the remaining half of the Parisi formula.
In this section, we shall apply Guerra’s comparison scheme (see the recent accounts by
Aizenman et al. (2007); Guerra (2005); Talagrand (2007a)) to the SK model with multidimen-
sional spins. However, we shall use also the ideas (and the language) of Aizenman et al. (2003).
In particular, we shall use the same local comparison functional (3.88) as in the AS2scheme,
see (4.2). The section contains the proofs of the upper (4.14) and lower (4.16) bounds on the free
energy without Assumption 3.1.2. The proofs use the GREM-like Gaussian processes, RPCs as
82 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
in the AS2scheme. We also obtain an analytic representation of the remainder term (which is
an artifact of this scheme) using the properties of the Bolthausen-Sznitman coalescent.
4.1.1 Multidimensional Guerra’s scheme
Given n∈
N
, assume that xand Qsatisfy (3.7) and (3.6), respectively. Recall the definitions of
the Gaussian processes Xand Awhich satisfy (0.8) and (3.15), respectively. We consider, for
t∈[0;1], the following interpolating Hamiltonian on the configuration space ΣN×A
Ht(σ,α):=√tX(σ)+ √1−tA(σ,α).(4.1)
Given U⊂Sym+(d), the Hamiltonian (4.1) in the usual way induces the following local free
energy
ϕN(t,x,Q,U):=ΦN(x,U)[Ht],(4.2)
where we use the same local comparison functional (3.88) as in the AS2scheme. Using (3.4),
we obtain then
ϕ(0,x,Q,U) = ΦN(x,U)[A]and ϕ(1,x,Q,U) = ΦN(x,U)[X] = pN(U).
Now, we are going to disintegrate the Gibbs measure defined on U×Ainto two Gibbs mea-
sures acting on Uand Aseparately. For this purpose we define the correspondent (random)
local free energy on Uas follows
ψ(t,x,Q,α,U):=logZΣN(U)
exphβ√NHt(σ,α)idµ⊗N(σ).(4.3)
For α∈A, we can define the (random) local Gibbs measure G(t,Q,α,U)∈M1(ΣN)by
demanding that the following holds
dG(t,x,Q,α,U)
dµ⊗N(σ):=
1
ΣN(U)(σ)exphβ√NHt(σ,α)−ψ(t,x,Q,U,α)i.
Let us define a certain reweighting of the RPC ξwith the help of (4.3). We define the random
point process {˜
ξ}α∈Ain the following way
˜
ξ(α):=ξ(α)exp(ψ(t,x,Q,U,α)).
We also define the normalisation operation N:Mf(A)→M1(A)as
N(ξ)(α):=ξ(α)
∑α0∈Aξ(α0).
We introduce the local Gibbs measure G(t,x,Q,U)∈M1(U×A), for any V⊂U×A, as
follows
G(t,x,Q,U)[V]:=∑
α∈An
N(˜
ξ)(α)G(t,x,Q,α,U)[V].(4.4)
Finally, we introduce, what shall call Guerra’s remainder term:
R(t,x,Q,U):=−β2
2
E
G(t,x,Q,U)⊗G(t,x,Q,U)kR(σ1,σ2)−Q(α1,α2)k2
F.(4.5)
Note that (4.5) coincides with (3.18) after substituting (3.16) with (3.21).
4.1 Guerra’s comparison scheme 83
4.1.2 Local comparison
The results of Section 3.4 can be straightforwardly generalised to the comparison scheme based
on (4.1). Given ε,δ>0 and Λ∈Sym(d), define
V(Λ,U,ε,δ):={U0∈Sym(d):kU0−UkF<ε,hU0−U,Λi<δ}.(4.6)
We now specialise to the case U=ΣN(V(Λ,U,ε,δ)).
Lemma 4.1.1. We have
∂
∂tϕN(t,x,Q,V(Λ,U,ε,δ)) =R(t,x,Q,ΣN(A(Λ,U,ε,δ)))
−β2
2
n
∑
k=1
xkkQ(k+1)k2
F−kQ(k)k2
F+O(ε).(4.7)
Proof. This is an immediate consequence of Proposition 3.2.5. Indeed, recalling that Q(α1,α1) =
U, and setting U:=Σ(B(U,ε)), we have
∂
∂tϕ(t,x,Q,U)
=β2
2
E
G(t,x,Q,U)⊗G(t,x,Q,U)kR(σ1,σ1)−Uk2
F−kR(σ1,σ2)−Q(α1,α2)k2
F
−kUk2
F−kQ(α1,α2)k2
F
=−β2
2
E
G(t,x,Q,U)⊗G(t,x,Q,U)kR(σ1,σ2)−Q(α1,α2)k2
F
−β2
2
E
G(t,x,Q,U)⊗G(t,x,Q,U)kUk2
F−kQ(α1,α2)k2
F+O(ε).(4.8)
Using the Proposition 2.2.1, we get
β2
2
E
G(t,x,Q,U)⊗G(t,x,Q,U)kUk2
F−kQ(α1,α2)k2
F
=β2
2
E
N(ξ)⊗N(ξ)
n
∑
k=qL(α(1),α(2))kQ(k+1)k2
F−kQ(k)k2
F
=β2
2
n
∑
k=1kQ(k+1)k2
F−kQ(k)k2
F
E
hN(ξ)⊗N(ξ){k≥qL(α(1),α(2))}i
=β2
2
n
∑
k=1
xkkQ(k+1)k2
F−kQ(k)k2
F.(4.9)
Combining (4.8) and (4.9), we get (4.7)
ut
Lemma 4.1.2. We have
pN(ΣN(B(U,ε))) =ΦN(x,ΣN(B(U,ε)))[A]−β2
2
n
∑
k=1
xkkQ(k+1)k2
F−kQ(k)k2
F
+Z1
0
R(t,x,Q,ΣN(B(U,ε))dt+O(ε).(4.10)
84 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
Remark 4.1.1. Note that the above lemma also holds if we substitute B(U,ε)with the smaller
set V(Λ,U,ε,δ).
Proof. The claim follows from (4.7) by integration. ut
Proposition 4.1.1. There exists C =C(Σ,µ)>0such that, for all U ∈Sym+(d)as above, and
all ε,δ>0, there exists an δ-minimal Lagrange multiplier Λ=Λ(U,ε,δ)∈Sym(d)in (3.11)
such that, for all t ∈[0;1], and all (x,Q), we have
pN(ΣN(V(Λ,U,ε,δ))) ≤inf
Λ∈Sym(d)f(x,Q,U,Λ)+C(ε+δ)(4.11)
and
lim
N↑+∞pN(ΣN(B(U,ε))) ≥inf
Λ∈Sym(d)f(x,Q,U,Λ)+ lim
N↑+∞Z1
0
R(t,x,Q,ΣN(B(U,ε)))dt
−C(ε+δ).(4.12)
Remark 4.1.2. The following upper bound also holds true. There exists C =C(Σ,µ)>0, such
that, for any Λ∈Sym(d),
pN(ΣN(B(U,ε))) ≤f(x,Q,U,Λ)+CkΛkFε.(4.13)
Proof. The result follows from Lemma 4.1.2 by the same arguments as in the proofs of Theo-
rems 3.1.1 and 3.1.2. ut
4.1.3 Free energy upper and lower bounds
Similarly to the quenched LDP bounds for the AS2scheme in the SK model with multidimen-
sional spins (see Section 3.3), we get the quenched LDP bounds for Guerra’s scheme in the
same model without Assumption 3.1.2 on Q.
Recall the definition of the local Parisi functional f(3.11).
Theorem 4.1.1. For any closed set V⊂Sym(d), we have
p(V)≤sup
U∈V∩U
inf
(x,Q,Λ)f(x,Q,Λ,U),(4.14)
where the infimum runs over all x satisfying (3.7), all Qsatisfying (3.6) and all Λ∈Sym(d).
Proof. The proof is identical to the one of Theorem 3.1.1. ut
Define the local limiting Guerra remainder term R(x,Q,U)as follows
R(x,Q,U):=−lim
ε↓+0lim
N↑+∞Z1
0
R(t,ΣN(B(U,ε)))dt≤0.(4.15)
The existence of the limits in (4.15) is proved similar to the case of the AS2scheme, see the
proof of Theorem 3.1.2.
4.1 Guerra’s comparison scheme 85
Theorem 4.1.2. For any open set V⊂Sym(d), we have
p(V)≥sup
U∈V∩U
inf
(x,Q,Λ)[f(x,Q,Λ,U)+R(x,Q,U)],(4.16)
where the infimum runs over all x satisfying (3.7); all Qsatisfying (3.6) and all Λ∈Sym(d).
Proof. The proof is identical to the one of Theorem 3.1.2. The only new ingredient is Lemma 4.1.1
needed to recover Guerra’s remainder term (4.5). ut
4.1.4 The filtered d-dimensional GREM
Given U∈Sym+(d)non-negative definite, denote by Q(U,d)the set of all c`
adl`
ag (right con-
tinuous with left limits) Sym+(d)-valued non-decreasing paths which end in matrix U, i.e.,
Q(U,d):={ρ:[0;1]→Sym+(d)|ρ(0) = 0;ρ(1) = U;ρ(t)ρ(s),for t≤s;ρis c´
adl´
ag}.
(4.17)
Definition (4.17) is a multidimensional generalisation of (1.26). Define the natural inverse ρ−1:
Imρ→[0;1]as
ρ−1(Q):=inf{t∈[0;1]|ρ(t)Q},
where Imρ:=ρ([0;1]). Let x:=ρ−1◦ρ∈Q(1,1).
Let also Q0(U,d)⊂Q(U,d)be the space of all piece-wise constant paths in Q(U,d)with
finite (but arbitrary) number of jumps with an additional requirement that they have a jump at
x=1. Given some ρ∈Q0(U,d), we enumerate its jumps and define the finite collection of
matrices {Q(k)}n+1
k=0:=Imρ⊂
R
d. This implies that there exist {xk}n+1
k=0⊂
R
such that
0=:x0<x1< ... < xn<xn+1:=1,
0=:Q(0)Q(1)Q(2)···Q(n+1):=U,
where ρ(xk) = Q(k). Let us associate to ρ∈Q0(U,d)a new path ˜
ρ∈Q(U,d)which is obtained
by the linear interpolation of the path ρ. Namely, let
˜
ρ(t):=Q(k)+(Q(k+1)−Q(k))t−xk
xk+1−xk
,t∈[xk;xk+1).
Let g:
R
d→
R
be a function satisfying Assumption 4.1.1. Let us introduce the filtered d-
dimensional GREM process W. Let
W:=n{Wk(t,[α]k)}t∈
R
+:α∈A,k∈[0;n]∩
N
o
be the collection of independent (for different αand k)
R
d-valued correlated Brownian motions
satisfying
Wk(t,[α]k)∼(Q(k+1)−Q(k))1/2Wt−xk
xk+1−xk,
86 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
where {W(t)}t∈
R
+is the standard (uncorrelated)
R
d-valued Brownian motion. Now, for k∈
[0;n]∩
N
, we define the
R
d-valued process {Y(t,α)|α∈A,t∈[0;1]}by
Y(t,α):=
n
∑
k=0
1
[xk;1](t)Wk(t∧xk+1,[α]k).
Lemma 4.1.3. For α(1),α(2)∈A, we have
CovhY(t1,α(1)),Y(t2,α(2))i=˜
ρt1∧t2∧xqL(α(1),α(2)).
Proof. The proof is straightforward. ut
Assumption 4.1.1. Suppose that the function g :
R
d→
R
satisfies g ∈C(2)(
R
d)and, for any
c>0, we have R
R
dexpg(y)−ckyk2
2dy<∞and also
sup
y∈
R
dk∇g(y)k2+k∇2g(y)k2<+∞,(4.18)
where ∇2g(y)denotes the matrix of second derivatives of the function g at y ∈
R
d.
Assume gsatisfies the above assumption. Let f:=fρ:[0;1]×
R
d→
R
be the function satis-
fying the following (backward) recursive definition
f(t,y):=(g(y),t=1,
1
xklog
E
expxkf(xk+1,y+Y(xk+1,α)−Y(t,α)),t∈[xk;xk+1),(4.19)
where k∈[0;n]∩
N
,α∈Ais arbitrary and fixed.
Remark 4.1.3. It is easy to recognise that the definition of f is a continuous “algorithmisation”
of (3.10). Namely, Xk(x,Q,Λ,U) = f(xk,0), where
f(1,y) = g(y):=logZΣexp√2βhy,σi+hΛσ,σidµ(σ).(4.20)
4.1.5 A computation of the remainder term
Recall the equivalence relation (2.27). In words, the equivalence i∼
kjmeans that the atoms of
the RPC ξwith ranks iand jhave the same ancestors up to the k-th generation. Varying the kin
(2.27), we get a family of equivalences on
N
which possesses important Markovian properties,
see Bolthausen & Sznitman (1998).
Lemma 4.1.4. For all k ∈[0;n−1]∩
N
, we have
E
∑
i∼
kj
i
k+1j
N(ξ)(i)N(ξ)(j)
=xk+1−xk,(4.21)
and also
E
h∑
i
N(ξ)(i)2i=1−xn.(4.22)
4.1 Guerra’s comparison scheme 87
Proof. 1. To prove (4.21) we notice that
E
∑
i∼
kj
i
k+1j
N(ξ)(i)N(ξ)(j)
=
E
∑
i
k+1j
N(ξ)(i)N(ξ)(j)−∑
i
kj
N(ξ)(i)N(ξ)(j)
=xk+1−xk,
where the last equality is due to Proposition 2.2.1.
2. Similarly, (4.22) follows from the following observation
E
"∑
i
N2(ξ)(i)#=
E
∑
i,j
N(ξ)(i)N(ξ)(j)−∑
i
nj
N(ξ)(i)N(ξ)(j)
=1−xn,
where the last equality is due to Proposition 2.2.1.
ut
Note that, using the above notations, we readily have
A(σ,α)∼2
N1/2N
∑
i=1hY(i)(1,α),σii,
where {Y(i):={Y(i)(1,α)}α∈A}N
i=1are i.i.d. copies of {Y(1,α)}α∈A. Consider the following
weights
˜
ξ(t)(α):=ξ(α)exp(f(t,Y(t,α))).
As in (Bolthausen & Sznitman, 1998), the above weights induce the permutation ˜
π(t):
N
→A
such that, for all i∈
N
, the following holds
˜
ξ(t)(˜
π(t)(i)) >˜
ξ(t)(˜
π(t)(i+1)).(4.23)
In what follows, we shall use the short-hand notations ˜
ξ(t)(i):=˜
ξ(t)(˜
π(t)(i)),˜
Y(t)(s,i):=
Y(s,˜
π(t)(i)) and ˜
Q(t):={˜
Q(t)(i,j):=Q(˜
π(t)(i),˜
π(t)(j))}i,j∈
N
.
Theorem 4.1.3. Given a discrete order parameter x ∈Q0(1,1), we have
1. Independence #1. The normalised RPC point process N(ξ)is independent from the corre-
sponding randomised limiting GREM overlaps q.
2. Independence #2. The reordered filtered limiting GREM ˜
Y is independent from the corre-
sponding reordered weights ˜
ξ.
3. The reordering change of measure. Given I b
N
, let νI(·|Q)be the joint distribution of
{Y(1,i)}i∈I, and ˜
νI(·|Q)be the joint distribution of {˜
Y(1)(1,i)}i∈Iboth conditional on Q.
Then
88 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
d˜
νI(·|Q)
dνI(·|Q)=
n
∏
k=0
∏
i∈I/∼
kexpxkf(xk+1,Y(xk+1,i))−fk(xkY(xk,i)),(4.24)
where the innermost product in the previous formula is taken over all equivalence classes
on the index set I induced by the equivalence ∼
k.
4. The averaging property. For all s,t∈[0;1], we have
nξ(t)(α)oα∈A,˜
Q(t)∼nξ(s)(α)oα∈A,˜
Q(s).(4.25)
Proof. The proof is the same as in the case of the one-dimensional SK model, see Arguin
(2007); Bolthausen & Sznitman (1998). ut
Keeping in mind (4.24), we define, for k∈[0;n−1]∩
N
, the following random variables
Tk(α):=expxkf(xk+1,Y(xk+1,α))−f(xk,Y(xk,α)).
Given k∈[1;n]∩
N
, assume that α(1),α(2)∈Asatisfy qL(α(1),α(2)) = k. We introduce, for no-
tational convenience, the (random) measure µk(t,U)– an element of M1(ΣN)– by demanding
the following
µk(t,U)[g]:=
E
T1(α1)···Tk(α1)Tk+1(α1)Tk+1(α2)···Tn(α1)Tn(α2)
G(t,α(1),U)⊗G(t,α(2),U)[g]i,(4.26)
where g:U2→
R
is an arbitrary measurable function such that (4.26) is finite. Using this
notation, we can state the following lemma.
Lemma 4.1.5. For any i,j∈
N
, satisfying i ∼
kj, i
k+1j, we have
E
G(t,i,U)⊗G(t,j,U)kR(σ1,σ2)−Q(i,j)k2
F=µk(t,U)hkR(σ1,σ2)−Q(k)k2
Fi.
(4.27)
Proof. This is a direct consequence of (4.24) and the fact that under the assumptions of the
theorem Q(i,j) = Q(k).
ut
Remark 4.1.4. It is obvious from the previous theorem that µkis a probability measure.
The main result of this subsection is an “analytic projection” of the probabilistic RPC repre-
sentation which integrates out the dependence on the RPC. Comparing to (3.18), it has a more
analytic flavor which will be exploited in the remainder estimates (Section 4.3). This is also a
drawback in some sense, since the initial beauty of the RPCs is lost.
Theorem 4.1.4. In the case of Guerra’s interpolation (3.21), we have
R(t,x,Q,ΣN(B(U,ε))) =1
2
n−1
∑
k=0
(xk+1−xk)µk(t,ΣN(B(U,ε)))hkR(σ1,σ2)−Q(k)k2
Fi
+O(ε)+O(1−xn),(4.28)
as ε→0and xn→1.
4.2 The Parisi functional in terms of differential equations 89
Proof. Recalling (4.5) and (4.4), we write
R(t,x,Q,Σ(U,ε)) = β2
2
E
h∑
i,j
N(˜
ξ)(i)N(˜
ξ)(j)
×G(t,x,Q,i,U)⊗G(t,x,Q,j,U)hkR(σ1,σ2)−Q(i,j)k2
Fii.
Using the Theorem 4.1.3, we arrive to
R(t,x,Q,Σ(U,ε)) = β2
2∑
i,j
E
hN(˜
ξ)(i)N(˜
ξ)(j)i
×
E
G(t,x,Q,i,U)⊗G(t,x,Q,j,U)kR(σ1,σ2)−Q(i,j)k2
F.
(We can interchange the summation and expectation since all summands are non-negative.) The
averaging property (see Theorem 4.1.3) then gives
R(t,Σ(U,ε)) = β2
2∑
i,j
E
[N(ξ)(i)N(ξ)(j)]
E
G(t,i,U)⊗G(t,j,U)kR(σ1,σ2)−Q(i,j)k2
F.
(4.29)
For each k∈[1;n−1]∩
N
, we fix any indexes i0,i(k)
0,j(k)
0∈
N
such that i∼
kjand i
k+1j.
Rearranging the terms in (4.29), we get
R(t,Σ(U,ε)) = β2
2
n
∑
k=1
E
hG(t,i(k)
0,U)⊗G(t,j(k)
0,U)hkR(σ1,σ2)−Q(k)k2
Fii
×∑
i∼
kj
i
k+1j
E
[N(ξ)(i)N(ξ)(j)]
+β2
2
E
G(t,i0,U)⊗G(t,i0,U)kR(σ1,σ2)−Uk2
F∑
i
E
N(ξ)(i)2.
(4.30)
Finally, applying Lemmata 4.1.4 and 4.1.5 to (4.30), we arrive at (4.28).
ut
4.2 The Parisi functional in terms of differential equations
In this section, we study the properties of the multidimensional Parisi functional. We derive the
multidimensional version of the Parisi PDE. This allows to represent the Parisi functional as
a solution of a PDE evaluated at the origin. We also obtain a variational representation of the
Parisi functional in terms of a HJB equation for a linear problem of diffusion control. As a by-
product, we arrive at the strict convexity of the Parisi functional in 1-D which settles a problem
of uniqueness of the optimal Parisi order parameter posed by Panchenko (2005a); Talagrand
(2006c).
90 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
Lemma 4.2.1. Consider the function B :
R
d×
R
+→
R
defined as
B(y,t):=1
xlog
E
[exp{x f (y+z(t))}],
where f :
R
d→
R
satisfies Assumption 4.1.1 and {z(t)}t∈[0;1]is a Gaussian
R
d-valued process
with Cov[z(t)] :=Q(t)∈Sym(d)such that Q(t)u,vis differentiable, for all u,v. Then
∂tB(y,t) = 1
2
d
∑
u,v=1
˙
Qu,v(t)∂2
yuyvB(y,t)+x∂yuB(y,t)∂yvB(y,t),(t,y)∈(0;1)×
R
d.(4.31)
In particular, the function B is differentiable with respect to the t-variable on (0;1)and C2(
R
d)
with respect to the y-variable.
Proof. Denote Z:=
E
hex f (y+z(t))i. By (Aizenman et al., 2007, Lemma A.1), we have
∂tB(y,t) = 1
2x 1
Z
E
hd
∑
u,v=1
˙
Qu,v(t)∂2
zuzvex f (z)|z=y+z(t)i!.
A straightforward calculation then gives
∂tB(y,t) = 1
2x 1
Z
E
hd
∑
u,v=1
˙
Qu,v(t)x2∂zuf(z)∂zvf(z)+x∂2
zuzvf(z)ex f (z)|z=y+z(t)i!.(4.32)
We also have
∂yuB(y,t) = 1
xZ
E
hxex f (z)∂zuf(z)|z=y+z(t)i,(4.33)
and
∂2
yuyvB(y,t) = 1
x1
Z
E
hex f (z)x2∂zuf(z)∂zvf(z)+∂2
zuzvf(z)|z=y+z(t)i
−1
Z2
E
hxex f (z)∂zuf(z)|z=y+z(t)i
E
hxex f (z)∂zvf(z)|z=y+z(t)i.(4.34)
Combining (4.32), (4.33) and (4.34), we get (4.31). ut
Proposition 4.2.1. Denote D :=Sn
k=0(xk;xk+1). The function f =fρdefined in (4.19) satisfies
the final-value problem for the controlled semi-linear parabolic Parisi-type PDE
∂tf(y,t)+ 1
2∑d
u,v=1d
dt˜
ρu,v(t)∂2
yuyvf(y,t)+x(t)∂yuf(y,t)∂yvf(y,t)=0,(t,y)∈D×
R
d,
f(1,y) = g(y),y∈
R
d,
f(y,xk−0) = f(y,xk+0),k∈[1;n]∩
N
,y∈
R
d.
(4.35)
Note that d
dt˜
ρ(t) = Q(k+1)−Q(k)
xk+1−xk, for t ∈(xk;xk+1).
4.2 The Parisi functional in terms of differential equations 91
Proof. A successive application of Lemma 4.2.1 to (4.19) on the intervals Dstarting from (xn;1)
gives (4.35). ut
Remark 4.2.1. Note that a straightforward inspection of (4.19), using (4.32),(4.33) and (4.34),
shows that the function f defined in (4.19) is C1(D)∩C([0;1]) with respect to the t-variable and
C2(
R
d)with respect to the y-variable.
Lemma 4.2.2. Given ρ∈Q0(U,d), the function (4.19) satisfies the following:
fρ(0,0) =
E
hlog ∑
α∈A
ξ(α)exp{g(Y(1,α))}i.(4.36)
Proof. This is an immediate consequence of the RPC averaging property (4.25). ut
Lemma 4.2.3.
1. Given k ∈[1;n]∩
N
and a non-negative definite matrix Q ∈Sym(d), we have
∂Q(k) Qfρ(0,0) = −1
2(xk−xk−1)
E
[hQ,Mi],(4.37)
where M ∈
R
d×dis defined as
Mu,v:=T1(α(1))···Tk(α(1))Tk+1(α(1))Tk+1(α(2))···Tn(α(1))Tn(α(2))
∂zug(z)|z=Y(1,α(1))∂zvg(z)|z=Y(1,α(2))
with qL(α(1),α(2)) = k. Moreover, (4.37) does not depend on the choice of α(1),α(2)∈A
but only on k.
2. Given a non-negative definite matrix Q ∈Sym(d), we have
∂U Qfρ(0,0) = 1
2
E
hQ,M0i,(4.38)
where M0∈Sym(d)is satisfies
M0
u,v=T1(α)···Tn(α)∂2
zuzvg(z)+∂zug(z)∂zvg(z)z=Y(1,α)+O(1−xn),
as xn→1. Note that (4.38) obviously does not depend on the choice of α∈A.
Proof. Applying (Aizenman et al., 2007, Lemma A.1) to (4.36), we obtain
∂s
E
hlog ∑
α∈A
exp{g(Y(1,α))}Q(k)=Q(k)+sQi
=1
2
E
hd
∑
u,v=1
N(˜
ξ)⊗N(˜
ξ)h∂sQ(α(1),α(2))u,v|Q(k)=Q(k)+sQ
n
1
α(1)=α(2)(α(1),α(2))∂2
zuzvg(z)+∂zug(z)∂zvg(z)z=Y(1,α(1))
−∂zug(z)|z=Y(1,α(1))∂zvg(z)|z=Y(1,α(2))oQ(k)=Q(k)+sQii.
Note that
∂sQ(α(1),α(2))u,v|Q(k)=Q(k)+sQ=(Qu,v,qL(α(1),α(2)) = k,
0,qL(α(1),α(2))6=k.
92 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
1. Define M(α(1),α(2))∈
R
d×das
M(α(1),α(2))u,v:=∂zug(z)|z=Y(1,α(1))∂zvg(z)|z=Y(1,α(2)).
Hence, we arrive at
∂Q(k) Qfρ(0,0) = −1
2
E
h∑
α(1)α(2)∈A
1
qL(α(1),α(2))=kξ(α(1))ξ(α(2))(α(1),α(2))hQ,M(α(1),α(2))ii.
The proof is concluded similarly to the proof of Theorem 4.1.4 by using the properties of
the RPC (Theorem 4.1.3 and Lemma 4.1.4).
2. The proof is the same as in (1).
ut
The following is a multidimensional version of (Talagrand, 2006b, Lemma 4.3).
Lemma 4.2.4. For any α∈A, we have
1.
∂xkfρ(0,0)|xk=xk−1=1
xk−1
E
hT1(α)···Tk−2(α)Tk−1(α)|xk=xk−1
E
hf(xk+1,Y(xk+1,α))Tk(α)|xk=xk−1i−f(xk,Y(xk,α))i.
2. Let M ∈Sym(d)with Mu,v:=∂zuf(xk,Y(xk,α))∂zvf(xk,Y(xk,α)), then
∂2
Q(k) Q,xk
fρ(0,0) = 1
2
E
hT1(α)···Tk−2(α)hQ,Mii.
Proof. This proof is the same as in (Talagrand, 2006b). ut
We now generalise the PDE (4.35). Given a piece-wise continuous x∈Q(1,1)and Q∈
Q(U,d), consider the following terminal value problem
(∂tf+1
2h˙
Q,∇2fi+xh˙
Q∇f,∇fi=0,(y,t)∈
R
d×(0,1),
f(y,1) = g(y).(4.39)
We say that f∈C([0;1]×
R
d→
R
)is a piece-wise viscosity solution of (4.39), if there exists
the partition of the unit segment 0 =:x0<x1< ... < xn+1:=1 such that, for each k∈[0,n]∩
N
,
f:(xk;xk+1)×
R
d→
R
is a viscosity solution (see, e.g., Briand & Hu (2007)) of
∂tf+1
2h˙
Q,∇2fi+xh˙
Q∇f,∇fi=0,(y,t)∈
R
d×(xk,xk+1),
f(y,xk+1+0) = f(y,xk+1−0),
f(y,1) = g(y).
Proposition 4.2.2. For any ρ(1),ρ(2)∈Q0(U,d), we have
|fρ(1)(0,0)−fρ(2)(0,0)|≤ C
2Z1
0kρ(1)(t)−ρ(2)(t)kFdt,
where C =C(Σ):=
E
kMkF.
4.2 The Parisi functional in terms of differential equations 93
Proof. This is an adaptation of the proof of (Talagrand, 2006c, Theorem 3.1) to the multidimen-
sional case. Assume without loss of generality that the paths ρ(1)and ρ(2)have same jump times
{xk}n+1
k=0. Denote the corresponding overlap matrices as {Q(1,k)}n+1
k=0and {Q(2,k)}n+1
k=0. Given s∈
[0;1], define the new path ρ(s)∈Q0(U,d)by assuming that it has the same jump times {xk}n+1
k=0
as the paths ρ(1),ρ(2)and defining its overlap matrices as Q(k)(s):=sQ(1,k)+(1−s)Q(2,k). On
the one hand, we readily have
Z1
0kρ(1)(t)−ρ(2)(t)kFdt=
n
∑
k=1
(xk−xk−1)kQ(1,k)−Q(2,k)kF.
On the other hand, using Lemma 4.2.3, we have
|∂sfρ(s)(0,0)|≤ C
2
n
∑
k=1
(xk−xk−1)kQ(1,k)−Q(2,k)kF.
Finally, we have
|fρ(1)(0,0)−fρ(2)(0,0)|≤Z1
0|∂sfρ(s)(0,0)|ds.
Combining the last three formulae, we get the theorem. ut
Remark 4.2.2. Note that using the same argument and notations as in the previous theorem we
get that, for any (y,t)∈
R
d×[0;1],
|fρ(1)(y,t)−fρ(2)(y,t)|≤ C(Σ)
2Z1
tkρ(1)(s)−ρ(2)(s)kFds.
Remark 4.2.3. Note that we can associate to each ρ∈Q(U,d)aSym+(d)-valued countably
additive vector measure νρ∈M([0;1],Sym+(d)) by the following standard procedure. Given
[a;b)⊂[0;1], define
νρ([a;b)) :=ρ(b)−ρ(a)
and then extend the measure, e.g., to all Borell subsets of [0;1].
Theorem 4.2.1. Given U ∈Sym+(d), we have
1. The set Q(U,d)is compact under the topology induced by the following norm
kρk:=Z1
0kρ(t)kFdt,ρ∈Q(U,d).(4.40)
2. The functional Q0(U,d)3ρ7→ fρ(0,0)is Lipschitzian and can be uniquely extended by
continuity to the whole Q(U,d).
Proof. 1. The topology induced by the norm (4.40) coincides with the topology of weak con-
vergence of the above-defined vector measures. Since Q(U,d)is a bounded set, it is compact
in the weak topology.
2. This is an immediate consequence of Proposition 4.2.2.
ut
94 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
In the next result, we summarise some results on the PDE (4.39) for the non-discrete param-
eters, cf. Proposition 4.2.1.
Theorem 4.2.2.
1. Existence. Assume that Q is in Q(U,d)and is piece-wise C(1). Assume also that x is in
Q(1,1)and is piece-wise continuous. Then the terminal value problem (4.39) has a unique
continuous, piece-wise viscosity solution fQ,x∈C([0;1]×
R
d).
2. Monotonicity with respect to x.Assume Q ∈Q(U,d). Assume also that x(1),x(2)∈Q(1,1)
are such that x(1)(t)≤x(2)(t), almost everywhere for t ∈[0;1]. Let fQ,x(1)and fQ,x(2)be the
corresponding solutions of (4.39). Then fQ,x(1)≤fQ,x(2).
3. Monotonicity with respect to g.Assume g1,g2:
R
d→
R
satisfy Assumption 4.1.1 and also
g1≤g2almost everywhere. Let fg1,fg2:
R
d×[0;1]→
R
be the corresponding solutions of
(4.39) with g =g1, g =g2, respectively. Then fg1≤fg2.
Proof. 1. Due to the assumptions, the diffusion matrix ˙
Q(t) = ˙
ρ(t)in (4.39) is non-negative
definite. Applying (Briand & Hu, 2007, Proposition 8) to the PDE (4.39) successively on
the intervals [xk;xk+1), where the ˙
ρis continuous, gives the existence of the solutions in
viscosity sense and, moreover, gives their continuity. Uniqueness is ensured by (Da Lio &
Ley, 2006, Theorem 1.1).
2. By the approximation argument (cf. Theorem 4.2.1), it is enough to assume that x(1),x(2)∈
Q0(1,1)and Q∈Q0(U,d). Then Proposition 4.2.1 gives the existence of the corresponding
piece-wise classical solutions of (4.39): fQ,x(1),fQ,x(2). These solutions are obviously also
the (unique) piece-wise viscosity solutions of (4.39). The comparison result (Briand & Hu,
2007, Theorem 5) and the non-linear Feynman-Kac formula (Briand & Hu, 2007, Proposi-
tion 8 ) give then the claim.
3. This can be seen either from the representation (4.36) and an approximation argument, or
exactly as in (2) by invoking the results of Briand & Hu (2007).
ut
4.2.1 The Parisi functional
We consider now a specific terminal condition in the system (4.35) given in (4.20).
Given ρ∈Q(U,d), let fρ:[0;1]×
R
d→
R
be the value of (the continuous extension onto
Q(U,d)of) the solution of (4.35) with the specific terminal condition given by (4.20). Fol-
lowing the ideas in the physical literature, we now define the Parisi functional P(β,ρ,Λ):
R
+×Q0(U,d)×Sym+(d)×Sym(d)→
R
in as
P(β,ρ,Λ):=fρ(0,0)−β2
2Z1
0
x(t)dkρ(t)k2
F−hU,Λi.(4.41)
The integral in (4.41) is understood in the usual Lebesgue-Stiltjes sense.
Remark 4.2.4. Note that the path integral term in (4.41) equals f (0,0), where f (t,y)is the
solution of (4.39) with the following boundary condition
g(y):=βhy,
1
i=β
d
∑
u=1
yu,y∈
R
d.
4.2 The Parisi functional in terms of differential equations 95
Obviously Q0(d)is dense in Q(d).
Theorem 4.2.3. We have
p(β)≤sup
U∈Sym+(d)
inf
ρ∈Q0(U,d)
Λ∈Sym(d)
P(β,ρ,Λ).(4.42)
Proof. The bound (4.42) is a straightforward consequence of Theorem 4.1.1. ut
4.2.2 On strict convexity of the Parisi functional and its variational representation
In this subsection, we derive a variational representation for Parisi’s functional. As a conse-
quence, for d=1, we prove that the functional is strictly convex with respect to the x∈Q(1,1),
if the terminal condition g(cf. (4.39)) is strictly convex and increasing. This result is related to
the problem of strict convexity of the Parisi functional in the case of the SK model.
Let W:={W(s)}s∈
R
+be the standard
R
d-valued Brownian motion and let {Ft}t∈
R
+be the
correspondent filtration. Define
U[t;T]:={u:[t;T]→
R
d|uis {Ft}t∈
R
+progressively measurable}.
Given u∈U[t;1],Q∈Q(U,d)and x∈Q(1,1), consider the following
R
d-valued and adapted
to {Ft}t∈
R
+diffusion
Y(Q,x,u,t,y)(s):=y−Zs
tx(s)˙
Q(s)1/2u(s)ds+Zs
t˙
Q(s)1/2dW(s),s∈[t;1].
Given some function g:
R
d→
R
satisfying Assumption 4.1.1, define fQ,x:
R
d×[0;1]→
R
as
fQ,x(y,t):=sup
u∈U[t;1]
E
g(Y(Q,x,u,t,y)(1))−1
2Z1
tku(s)k2
2ds.(4.43)
Proposition 4.2.3. Let d =1. If g is strictly convex and increasing, then the functional Q(1,1)3
x7→ fQ,xis strictly convex.
Proof. We have
Y(Q,x,u,t,y)(1) = y−Z1
tx(s)˙
Q(s)1/2u(s)ds+Z1
t˙
Q(s)1/2W(s).
By an approximation argument, it is enough to prove the strict convexity for the continuous
x1,x2∈Q(1,1)(x16=x2). For any γ∈(0;1), we have
Y(Q,γx1+(1−γ)x2,u,t,y)(1) = −Z1
tγx1+(1−γ)x2˙
Q(s)1/2u(s)ds+Z1
t˙
Q(s)1/2W(s)
<−γZ1
tx1˙
Q(s)1/2u(s)ds−(1−γ)Z1
tx2˙
Q(s)1/2u(s)ds
+Z1
t˙
Q(s)1/2W(s)
=γY(Q,x1,u,t,y)(1)+(1−γ)Y(Q,x2,u,t,y)(1),(4.44)
96 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
where the strict inequality above is due to the strict concavity of the square root function. The
strict convexity and monotonicity of gcombined with the representation (4.44) implies that
(4.43) is strictly convex as a function of x, since a supremum of a family of convex functions is
convex. ut
Proposition 4.2.4. Given a piece-wise continuous x ∈Q(1,1)and a Q ∈Q(U,d)which is
piece-wise in C1(0;1), the function fQ,x:
R
d×[0;1]→
R
defined by (4.43) is a unique, contin-
uous, piece-wise viscosity solution of the following terminal value problem
(∂tf+1
2h˙
Q,∇2fi+xh˙
Q∇f,∇fi=0,(y,t)∈
R
d×(0,1),
f(y,1) = g(y).
Proof. In a way similar to the proof of Theorem 4.2.2, we successively use (Da Lio & Ley,
2006, Theorem 2.1) on the intervals (xk;xk+1), where the data of the PDE are continuous. ut
Theorem 4.2.4. Assume d =1. Suppose also that g satisfies the assumptions of Proposi-
tion 4.2.3. For any u ∈
R
, the generalised Parisi functional given by (4.41) with fρ(0,0)corre-
sponding to the terminal condition g is strictly convex on Q(u,1). Consequently, there exists a
unique optimising order parameter.
Proof. In 1-D, we can choose the coordinates such that Q:=Ut, on [0;1]. Consequently, ˙
Q:=
U:=const on [0;1]. Hence, it is enough check the strict convexity with respect to x∈Q(1,1).
The result follows by approximation in the norm (4.40) of an arbitrary pair of different elements
of Q(U,d)by a pair of elements of Q0(U,d)and Propositions 4.2.1, 4.2.3 and 4.2.4. ut
Remark 4.2.5. Due to the monotonicity assumption on g, Theorem 4.2.4 does not cover the
case of the SK model, where the terminal value g is given by (4.20).
4.2.3 Simultaneous diagonalisation scenario
In the setups with highly symmetric state spaces ΣN(such as the spherical spin models of
Panchenko & Talagrand (2007b) or the Gaussian spin models, see Section 5.2 below), less
complex order parameter spaces as Q(U,d)suffice.
Given some orthogonal matrix O∈O(d), we briefly discuss the case ρ∈Qdiag(U,O,d),
where
Qdiag(U,O,d):={ρ∈Q(U,d)|for all t∈[0;1], the matrix Oρ(t)O∗is diagonal}.
The space Qdiag(U,O,d)is obviously isomorphic to the space of “paths” with the non-
decreasing coordinate functions in
R
d, starting from the origin and ending at u, i.e.,
¯
Q(u,d):={ρ:[0;1]→
R
d|¯
ρ(0) = 0; ¯
ρ(1) = u;¯
ρ(t)¯
ρ(s),for t≤s;¯
ρis c´
adl´
ag},
where u=OUO∗∈
R
d. The isomorphism is then given by
¯
Q(u,d)3¯
ρ7→OρO∗∈Qdiag(U,O,d).(4.45)
4.3 Remainder estimates 97
4.3 Remainder estimates
In this section, we partially extend Talagrand’s remainder estimates to the multidimensional set-
ting. Due to Proposition 4.1.1, to prove the validity of Parisi’s formula it is enough to show that
all the µkterms in (4.28) almost vanish for the almost optimal parameters of the optimisation
problem in (4.14). This can be done if the free energy of two coupled replicas of the system
(4.48) is strictly smaller than twice the free energy of the uncoupled single system (4.2), see
inequality (4.3.2). However, the systems involved in (4.3.2) are effectively at least as complex
as the SK model itself. In Section 4.3.2, we again apply Guerra’s scheme to obtain the upper
bounds on (4.48) in terms of the free energy of the corresponding comparison GREM-inspired
model. One might then hope that by a careful choice of the comparison model one can prove
inequality (4.3.2). In Sections 4.3.3 and 4.3.4, we formulate some conditions on the comparison
system which would suffice to get inequality (4.3.2), giving, hence, the conditional proof of the
Parisi formula, see Theorem 4.3.1.
4.3.1 A sufficient condition for µk-terms to vanish
In this subsection, we are going to establish a sufficient condition for the measures µkto vanish.
This condition states roughly the following. Whenever the free energy of a certain replicated
system uniformly in Nstrictly less then twice the free energy of the single system, the measure
µkvanishes in N→+∞limit (see Lemma 4.3.2).
Keeping in mind the definition of µk(cf. (4.26)) and of the Hamiltonian Ht(σ,α)(cf. (4.1)),
we define, for α(1),α(2)∈A(2),k, the corresponding replicated Hamiltonian as
H(2)
t(σ(1),σ(2),α(1),α(2)):=Ht(σ(1),α(1))+ Ht(σ(2),α(2)).(4.46)
Remark 4.3.1. We note here that the distribution of the Hamiltonian Ht(k,σ(1),σ(2))depends
only on k and not on the choice of the indices α(1),α(2)∈A(2),k.
Remark 4.3.2. The superscript (2)in (4.46) (and in what follows) indicates that the quantity is
related to the twice replicated objects.
Define
A(2),k:={(α(1),α(2))∈A2:qL(α(1),α(2)) = k}.
Additionally, for any V⊂Σ(B(U,ε))2and any suitable Gaussian process,
{F(σ(1),σ(2),α(1),α(2)):σ(1),σ(2)∈ΣN,α(1),α(2)∈A},
we define the local remainder comparison functional as
Φ(2),k,x
V[F]:=1
N
E
hlogZZVZZA(2),kexpnβ√NF(σ(1),σ(2),α(1),α(2))o
dµ⊗N(σ(1))dµ⊗N(σ(2))dξ(α(1))dξ(α(2))i.(4.47)
Define
ϕ(2)
N(k,t,x,Q,V):=Φ(2),k
VhH(2)
ti.(4.48)
98 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
Lemma 4.3.1. Recalling the definition (4.2), for any V⊂Σ(B(U,ε))2, we have
ϕ(2)
N(k,t,x,Q,V)≤ϕ(2)
N(k,t,x,Q,Σ(B(U,ε))2) = 2ϕN(t,x,Q,Σ(B(U,ε))).(4.49)
Proof. The first inequality in (4.49) is obvious, since the expression under the integral in (4.47)
is positive. The equality in (4.49) is an immediate consequence of the RPC averaging property
(4.25). ut
In what follows, we shall be looking for the sharper (in particular, strict) versions of the
inequality (4.49) because of the following observation due to Talagrand (2006b).
Lemma 4.3.2. Fix an arbitrary V⊂ΣN(B(U,ε))2. Suppose that, for some ε>0, the following
inequality holds
ϕ(2)
N(k,t,x,Q,V)≤2ϕN(t,x,Q,ΣN(B(U,ε)))−ε.(4.50)
Then, for some K >0, we have
µk(V)≤Kexp−N
K.
Proof. The proof is based on Theorem 3.2.2 and follows the lines of (Panchenko, 2005b,
Lemma 7). ut
4.3.2 Upper bounds on ϕ(2): Guerra’s scheme revisited
In this subsection, we shall develop a mechanism to obtain upper bounds on ϕ(2)defined in
(4.48). This will be achieved in the full analogy to Guerra’s scheme by using a suitable Gaussian
comparison system.
Given U∈Sym+(d), we say that V∈
R
d×dis an admissible mutual overlap matrix for U, if
U:=U V
V∗U∈Sym+(2d).(4.51)
Furthermore, define
V(U):={V∈
R
d×d:Vis an admissible mutual overlap matrix for U}.
Hereinafter without further notice we assume that U∈Sym+(2d)has the form (4.51), where
Vis some admissible mutual overlap matrix for U.
Let Q∈Q(U,2d). Let x:={xl∈[0;1]}n
l=1be the “jump times” of the path ρ. We assume
that the “times” are increasingly ordered, i.e.,
0=x0<x1< ... < xn<xn+1=1.
Consider the following collection of matrices
Q:={Ql:=Q(xl)⊂Sym+(2d)}n+1
l=0.
We obviously then have
4.3 Remainder estimates 99
0=Q(0)≺Q(1)≺...≺Q(n)≺Q(n+1)=U.(4.52)
Such a path Qinduces in the usual way the “doubled” GREM overlap kernel Q:={Q(α(1),α(2))∈
Sym+(2d)|α(1),α(2)∈An}, defined as
Q(α(1),α(2)):=Q(qL(α(1),α(2))).
We also need the d×dsubmatrices of the above overlap such that
Q(α(1),α(2)) = Q|11(α(1),α(2))Q|12(α(1),α(2))
Q|12(α(1),α(2))∗Q|22(α(1),α(2)).(4.53)
Remark 4.3.3. For σ(1)σ(2)∈ΣN, we shall use the notation σ(1)qσ(2)∈
R
2dNto denote the
vector obtained by the following concatenation of the vectors σ(1)and σ(2)
σ(1)qσ(2):=σ(1)
iσ(2)
i∈Σ×Σ⊂
R
2dN
i=1.
Let us observe that the process
X(2):=nX(2)(τ) = X(σ(1))+ X(σ(1))|τ=σ(1)qσ(2);σ(1),σ(2)∈ΣNo
is actually an instance of the 2d-dimensional Gaussian process defined in (0.8). Hence, it has
the following correlation structure, for τ1,τ2∈Σ(2)
N,
CovhX(2)(τ1),X(2)(τ2)i=kR(2)(τ1,τ2)k2
F.
The path ρinduces also the following two new (independent of everything before) comparison
process Y(2):=nY(2)(α)∈
R
2d|α∈Ano, with the following correlation structures
CovhY(2)(α(1)),Y(2)(α(2))i=Q(α(1),α(2))∈Sym+(d).
As usual, let {Y(2)
i}N
i=1be the independent copies of Y(2). For the purposes of new Guerra’s
scheme we define a GREM-like process (cf. (3.15))
A(2)={A(2)(τ,α):τ=σ(1)qσ(2);σ(1),σ(2)∈ΣN;α∈An}
as
A(2)(τ,α):=2
N1/2N
∑
i=1hY(2)
i(α),τii.
We fix some t∈[0;1]. We would now like to apply Guerra’s scheme to the comparison func-
tional (4.47) and the following two processes
nH(2)
t(σ(1),σ(2),α)oσ(1),σ(2)∈ΣN,α∈A,n√tA(2)(σ(1)qσ(2),α)oσ(1),σ(2)∈ΣN,α∈A.
100 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
These two processes are, respectively, the counterparts of the processes X(σ)and A(σ,α)in
Guerra’s scheme.
Consider a path e
Q∈Q0(U,d)with the following jumps
0=:e
Q(0)≺e
Q(1)≺...≺e
Q(n)≺e
Q(n+1).
Let e
A:=ne
A(σ,α):σ∈ΣN;α∈Anobe a Gaussian process (independent of all random ob-
jects around) with the following covariance structure
E
he
A(σ(1),α(1))e
A(σ(2),α(2))i=2hR(σ(1),σ(2)),e
Q(α(1),α(2))i.
For notational convenience, we introduce also the following process
e
A(2)(σ(1)qσ(2),α(1),α(2)):=e
A(σ(1),α(1))+ e
A(σ(2),α(2)).(4.54)
Recalling the replicated Hamiltonian (4.46) and following Guerra’s scheme, we introduce, for
s∈[0;1], the following interpolating Hamiltonian
H(2)
t,s(σ(1),σ(2),α(1),α(2)):=√stX(2)(σ(1)qσ(2))+p(1−s)tA(2)(σ(1)qσ(2),α(1))
+√1−te
A(2)(σ(1)qσ(2),α(1),α(2)).(4.55)
Given ε,δ>0 and L∈Sym(2d), define (cf. (4.6))
V(2)(L,U,ε,δ):={U0∈Sym+(2d):kU0−UkF<ε,hU0−U,Li<δ}.
We consider the following set of the local configurations
Σ(2)
N(L,U,ε,δ):=n(σ(1),σ(2))∈ΣN×ΣN:R(2)
N(σ(1)qσ(2),σ(1)qσ(2))∈V(2)(L,U,ε,δ)o.
(4.56)
Note that Σ(2)
N(L,U,ε,δ)⊂ΣN(B(U,ε))2. We consider also the RPC ζ=ζ(x)generated by the
vector xand, for any suitable Gaussian process
F:={F(σ(1),σ(2),α(1),α(2))|σ(1),σ(2)∈ΣN;α(1),α(2)∈An},
define the corresponding local comparison functional (cf. (4.47)) as follows
Φ(2),k,x
V[F]:=1
N
E
hlogZZVZZA(2),kexpnβ√NF(σ(1),σ(2),α(1),α(2))o
dµ⊗N(σ(1))dµ⊗N(σ(2))dζ(α(1))dζ(α(2))i.
Define the corresponding local free energy-like quantity as (cf. (4.2))
χ(s,t,k,x,Q,e
Q,Σ(2)
N(L,U,ε,δ)) :=Φ(2),k,x
Σ(2)
N(L,U,ε,δ)hH(2)
t,si.(4.57)
To lighten the notation, we indicate hereinafter only the dependence of χon s. Denote
Bx,Q:=tβ2
2
n
∑
l=1
xlkQ(l+1)k2
F−kQ(l)k2
F.
4.3 Remainder estimates 101
Lemma 4.3.3. There exists C =C(Σ)>0such that, for any Uas above, we have
∂
∂sχ(s,t,k,x,Q,e
Q,Σ(2)
N(L,U,ε,δ)) ≤−Bx,Q+Cε,(4.58)
Consequently,
ϕ(2)
N(k,t,x,Q,Σ(2)
N(L,U,ε,δ)) ≤Φ(2),k,x
Σ(2)
N(L,U,ε,δ)h√tA(2)(σ(1)qσ(2),α(1))
+√1−te
A(2)(σ(1)qσ(2),α(1),α(2))i−Bx,Q+Cε.(4.59)
Proof. The idea is the same as in the proof of Theorem 4.1.1 and is based on Proposition 3.2.5.
Since we are considering the localised free energy-like quantities (4.57), the variance terms
induced by the interpolation (4.55) in (3.34) cancel out (up to the correction O(ε)) and we are
left with the non-positive contribution of the covariance terms. ut
Given L∈Sym(2d), we consider the following stencil of the Legendre transform
e
Φ(2),k,x,L[F]:=−hL,Ui−Bx,Q+1
N
E
[logZZΣ2
NZZA(2),kexp{β√NF(σ(1),σ(2),α(1),α(2))
+hL(σ(1)qσ(2)),σ(1)qσ(2)i}
dµ⊗N(σ(1))dµ⊗N(σ(2))dζ(α(1))dζ(α(2))i.
(4.60)
Definition 4.3.1. Let F : Sym(2d)→
R
. Given δ>0, we call L(0)∈Sym(2d)δ-minimal for
F, if
F(Λ(0))≤inf
Λ∈Sym(2d)F(Λ)+δ.
Lemma 4.3.4. There exists C =C(Σ)>0such that, for all Uand Q∈Q0(U,2d)as above,
all ε,δ>0, there exists a δ-minimal Lagrange multiplier L=L(U,ε,δ)∈Sym(2d)for (4.60)
such that, for all k ∈[1;n]∩
N
, all t ∈[0;1], and all (x,Q), we have
ϕ(2)
N(k,t,x,Q,Σ(2)
N(L,U,ε,δ)) ≤inf
L∈Sym(2d)e
Φ(2),k,x,Lh√tA(2)(σ(1)qσ(2),α(1))
+√1−te
A(2)(σ(1)qσ(2),α(1),α(2))i
+C(ε+δ).(4.61)
Proof. The argument is the same as in the proof of Theorem 3.1.1. ut
Consider the family of matrices e
Q:=ne
Q(l)∈Sym+(2d)|l∈[0;n+1]∩
N
o, defined as
e
Q(l):=e
Q(l)e
Q(l)
e
Q(l)e
Q(l),(4.62)
for l∈[0;k]∩
N
, and as
102 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
e
Q(l):=e
Q(l)e
Q(k)
e
Q(k)e
Q(l),(4.63)
for l∈[k+1;n+1]∩
N
. Additionally we define, for l∈[0;n+1], the matrices
b
Q(l)(t):=tQ+(1−t)e
Q.
Let b
Z(l)∈
R
2d×2d, for l∈[0;x], be independent Gaussian vectors with
Covhb
Z(l)i=2β2b
Q(l+1)(t)−b
Q(l)(t).
Given b
y∈
R
2d,L∈Sym(2d), consider the random variable
X(2)
n+1(b
y,x,b
Q(t),L):=logZΣZΣexphb
y,σ(1)qσ(2)i+hL(σ(1)qσ(2)),σ(1)qσ(2)idµ(σ(1))dµ(σ(2)).
(4.64)
Define recursively, for l∈[n;0]∩
N
, the following quantities
X(2)
l(b
y,k,x,b
Q(t),L):=1
xl
log
E
b
Z(l)hexpxlX(2)
l+1(b
y+b
Z(l),k,x,b
Q(l)(t),L)i.(4.65)
Lemma 4.3.5. We have
e
Φ(2),k,x,Lh√tA(2)(σ(1)qσ(2),α(1))+√1−te
A(2)(σ(1)qσ(2),α(1),α(2))i
=−hL,Ui+X(2)
0(0,x,b
Q(l)(t),L).(4.66)
Proof. This is an immediate consequence of the RPC averaging property (4.25). ut
Proposition 4.3.1. Under the conditions of Lemma 4.3.4, we have
ϕ(2)
N(k,t,x,Q,Σ(2)
N(L,U,ε,δ)) ≤inf
L∈Sym(2d)−hL,Ui+X(2)
0(0,x,b
Q(t),L)−Bx,Q+C(ε+δ).
Remark 4.3.4. Similarly to (4.13), there exists C =C(Σ,µ)>0, such that, for any L∈
Sym(2d),
ϕ(2)
N(k,t,x,Q,Σ(2)
N(B(U,ε)) ≤−hL,Ui−Bx,Q+X(2)
0(0,x,b
Q(t),L)+CkLkFε.
Proof. Immediately follows from Lemmata 4.3.4 and 4.3.5. ut
4.3.3 Adjustment of the upper bounds on ϕ(2)
Proposition 3.2.1 implies that there exists r∈[1;n]∩
N
such that
kQ(r−1)k2
F<kVk2
F<kQ(r)k2
F.(4.67)
Assume r=k. (Other cases are similar or easier as shown for 1-D in Talagrand (2006b).) We
make the following tuning of the upper bounds of the previous subsection. Set n:=n+1. Let
w∈[xr−1/2;xr]. Define
4.3 Remainder estimates 103
xl:=xl(w):=
xl
2,l∈[0;k−1]∩
N
,
w,l=k,
xl,l∈[k+1;n+1]∩
N
.
(4.68)
Let
e
Q(l):=(Q(l),l∈[0;k−1]∩
N
,
Q(l−1),l∈[k;n+2]∩
N
.
Moreover, suppose Q:={Q(l)}n+2
l=0satisfy
kQ(l)k2
F=
4kQ(l)k2
F,l∈[0;k−1]∩
N
,
4kVk2
F,l=k,
2kQ(l−1)k2
F+kVk2
F,l∈[k+1;n+2]∩
N
.
(4.69)
Such Qexists due to (4.67). Moreover, if d≥2, then it is obviously non-unique.
Lemma 4.3.6. In the above setup, we have
Bx,Q:=tβ2n(w−xl−1)kQ(k)k2
F−kVk2
F+
n
∑
l=1
xlkQ(l+1)k2
F−kQ(l)k2
Fo.
Proof. The claim is a straightforward consequence of (4.68) and (4.69). ut
Define the matrix D(n+1)∈Sym+(2d)block-wise as
D(n+1)|11 :=β2t(U−Q(n+1)|11)+β2(1−t)(U−Q(n))+L|11,
D(n+1)|12 :=β2t(V−Q(n+1)|12)+L|12,
D(n+1)|21 :=β2t(V−Q(n+1)|12)∗+L|∗
12,
D(n+1)|22 :=β2t(U−Q(n+1)|22)+β2(1−t)(U−Q(n))+L|22.
Furthermore, we define
Sym+(2d)3e
D(n+1):=β2(U−Q(n))+Λ0
0β2(U−Q(n))+Λ.
Lemma 4.3.7. We have
X(2)
n+1(b
y,k,x,b
Q(t),L):=logZΣZΣexphb
y,σ(1)qσ(2)i+hD(n+1)(σ(1)qσ(2)),σ(1)qσ(2)i
×dµ(σ(1))dµ(σ(2)).
Proof. Since xn+2=1, the result follows from a straightforward calculation of the Gaussian
integrals in (4.65) for l=n+1. ut
Define
e
L:=Λ0
0Λ,e
U:=U Q(k)
Q(k)U.
104 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
Lemma 4.3.8. For any y ∈
R
d, l ∈[0;n+2]∩
N
, we have
X(2)
l(yqy,x(w),e
Q,e
L)|w=xk−1=(2Xl−1(y,x,Q,U,Λ),l∈[k;n+2]∩
N
,
2Xl(y,x,Q,U,Λ),l∈[0;k−1]∩
N
.
Proof. A straightforward (decreasing) induction argument on lgives the result. Indeed: for l=
n+2, an inspection of (4.64) and (3.8) immediately yields
X(2)
n+2(y(1)qy(2),x(w),e
Q,e
L) = Xn+1(y(1),x,Q,U,Λ)+Xn+1(y(2),x,Q,U,Λ),
where y(1),y(2)∈
R
d. Let b
Z(l)be a Gaussian 2d-dimensional vector with
Covhe
Z(l)i=2β2(e
Q(l+1)−e
Q(l)).
Define two Gaussian d-dimensional vectors e
Z(l),1and e
Z(l),2by demanding that
e
Z(l)=e
Z(l),1qe
Z(l),2.
Due to (4.62) and (4.63), the vectors e
Z(l),1and e
Z(l),2are independent, for l∈[k;n+1]. We have
e
Z(l),1∼e
Z(l),2, for l∈[0;k−1]. Assume that l∈[k;n+1]∩
N
and
X(2)
l+1(y(1)qy(2),x(w),e
Q,e
L) = Xl(y(1),x,Q,U,Λ)+Xl(y(2),x,Q,U,Λ).
By definition (4.64), we have
X(2)
l(y(1)qy(2),k,x,e
Q,e
L) = 1
xl
log
E
e
Z(l)hexpxlX(2)
l+1(y(1)qy(2)+b
Z(l),k,x,e
Q,L)i
=1
xl
log
E
e
Z(l)hexpnxlXl(y(1)+e
Z(l),1,x,Q,U,Λ)
+Xl(y(2)+e
Z(l),2,x,Q,U,Λ)oi
=Xl−1(y(1),x,Q,U,Λ)+Xl−1(y(2),x,Q,U,Λ).
By the construction and previous formula, for l=k−1, we have
X(2)
k−1(y(1)qy(2),k,x,e
Q,e
L)|w=xk−1=X(2)
k(y(1)qy(2),k,x,e
Q,e
L)
=Xk−1(y(1),x,Q,U,Λ)+Xk−1(y(2),x,Q,U,Λ).
Finally, for l∈[0;k−2], we recursively obtain
X(2)
l(y(1)qy(1),k,x,e
Q,e
L)|w=xk−1=1
xl
log
E
e
Z(l)hexpxlX(2)
l+1(y(1)qy(1)+b
Z(l),k,x,e
Q,L)|w=xk−1i
=2
xl
log
E
e
Z(l),1hexpnxl
2Xl+1(y(1)+e
Z(l),1,x,Q,U,Λ)
+Xl+1(y(1)+e
Z(l),1,x,Q,U,Λ)oi
=2Xl(y(1),x,Q,U,Λ).
ut
4.3 Remainder estimates 105
Remark 4.3.5. Motivated by Lemmata 4.3.2 and 4.3.8 (see also Section 4.3.4), we pose the
following problem. Is it true that, as in 1-D (see Panchenko (2005b); Talagrand (2006b)), there
exists Q∈Q0(U,2d)satisfying the assumption (4.69) such that the following inequality holds
inf
L∈Sym(2d)−hL,Ui+X(2)
0(0,x(w),b
Q(t),L)|w=xk−1
?
≤2 inf
Λ∈Sym(d)−hΛ,Ui+X0(0,x,Q,U,Λ)? (4.70)
Similar problems have at first been posed in Talagrand (2007b). The resolution of the above
problem seems to require more detailed information on the behaviour of the Parisi functional
(4.41) or, equivalently, of the solution of (4.39) as a function of Q ∈Q(U,d).
4.3.4 Talagrand’s a priori estimates
We start from defining a class of the almost optimal paths for the optimisation problem in (4.42).
Recall the following convenient definition from Panchenko (2005b).
Definition 4.3.2. Given U ∈Sym+(d), we shall call the triple (n,ρ∗,Λ∗)∈
N
×Q0
n(U,d)×
R
d
aθ-optimiser of the Parisi functional (4.41), if it satisfies the following two conditions
P(β,ρ∗,Λ∗)≤inf
ρ∈Q0(U,d)
Λ∈Sym(d)
P(β,ρ,Λ)+θ.(4.71)
P(β,ρ∗,Λ∗) = inf
ρ∈Q0
n(U,d)
Λ∈Sym(d)
P(β,ρ,Λ).(4.72)
Remark 4.3.6. It is obvious that for any θ>0such a θ-optimiser exists. The main convenient
feature of this definition (as pointed out in Talagrand (2006b)) is that n (the number of jumps of
ρ∗) is finite and fixed.
Recalling (4.11), we set
φ(x,Q,Λ)(t):=−hU,Λi−tβ2
2
n
∑
k=1
xkkQ(k+1)k2
F−kQ(k)k2
F+X0(x,Q,U,Λ).(4.73)
Under the following assumption (at first proposed in 1-D in Talagrand (2006b)), we shall ef-
fectively prove that remainder term almost vanishes on the θminimisers of (4.41), see Theo-
rem 4.3.1.
Assumption 4.3.1. Let U∈Sym+(2d)be defined by (4.51). We fix arbitrary t0∈[0;1),ε>0
and δ>0. There exists K =K(t0,ε,δ,U)>0,θ(t0,ε,δ,U)>0, and N0=N0(t0,ε,δ,U)∈
N
and L∗∈Sym(2d)with the following property:
If (n,ρ∗,Λ∗)is a θ-optimiser, for some θ∈(0;θ(t0,ε,δ,U)], then uniformly, for all t ∈
[0;t0), N >N0and all k ∈[1;n]∩
N
, we have
ϕ(2)
N(k,t,x∗,Q∗,Σ(2)
N(L∗,U,ε,δ)) ≤2φ(x∗,Q∗,Λ)(t)−1
KkQ∗(k)−Vk2
F+C(ε+δ).(4.74)
106 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
Remark 4.3.7. The validity of the above assumption for general a priori measures is an open
problem. However, in the particular case of the Gaussian a priori distribution the assumption
is indeed effectively satisfied. See Section 5.2 and Theorem 5.2.1, in particular. This gives a
complete proof of the Parisi formula for the case of Gaussian spins.
Remark 4.3.8. If the bound (4.70) holds then Lemma 4.3.6 with w =xr−1would imply that
ϕ(2)
N(k,t,Σ(2)
N(L∗,U,ε,δ)) ?
≤2φ(x∗,Q∗,Λ∗)(t)+C(ε+δ).(4.75)
The above inequality would then be a starting point for the a priori estimates in the spirit of
Talagrand (2006b) which might lead to the proof of Assumption 4.3.1.
4.3.5 Gronwall’s inequality and the Parisi formula
Theorem 4.3.1. Suppose Assumption 4.3.1 holds.
Then we have
lim
N↑+∞pN(β) = sup
U∈Sym+(d)
inf
ρ∈Q0(U,d)
Λ∈Sym(d)
P(β,ρ,Λ).
Proof. The proof follows the argument of Talagrand (2006b) (see also Panchenko (2005b)) with
the adaptations to the case of multidimensional spins. The main ingredients are the Gronwall
inequality and Lemma 4.3.2. Theorem 4.1.1 implies that
lim
N↑+∞pN(β)≤sup
U∈Sym+(d)
inf
ρ∈Q0(U,d)
Λ∈Sym(d)
P(β,ρ,Λ).
We now turn to the proof of the matching lower bound. As in the proof of Theorem 3.1.2, it is
enough to show that
lim
ε↓+0lim
N↑+∞ϕN(1,x,Q,B(U,ε)) ≥inf
ρ∈Q0(U,d)
Λ∈Sym(d)
P(β,ρ,Λ).(4.76)
1. We fix an arbitrary U∈Sym+(d). Fix also some t0∈[0;1). By Assumption 4.3.1, we can
find the corresponding θ(t0,V,U)>0 with the properties listed in the assumption. We pick
any θ∈(0;θ(t0,V,U)] and let (n,ρ∗,Λ∗)be a correspondent θ-optimiser. Note that, by
definition (4.73), we have
φ(x∗,Q∗,Λ∗)(1) = P(β,ρ∗,U,Λ∗)
and, by Definition 4.3.2,
|φ(x∗,Q∗,Λ∗)(1)−inf
ρ∈Q0(U,d)
Λ∈Sym(d)
P(β,ρ,U,Λ)|≤θ.(4.77)
2. We denote
∆N(t):=φ(x∗,Q∗,Λ∗)(t)−ϕN(t,x∗,Q∗,B(U,ε)).
4.3 Remainder estimates 107
Note that, due to (4.10), we obviously have
∆N(t)≥−Cε.(4.78)
Define
∆(t):=lim
N↑+∞∆N(t).
The definition (4.73) and Theorem 4.1.4 yield
d
dt∆N(t)≤1
2
n−1
∑
k=0
(xk+1−xk)µkhkRN(σ(1),σ(2))−Q(k)k2
Fi+Cε.(4.79)
3. Let us set D:=supσ∈Σkσk2. We note that, for any σ(1),σ(2)∈ΣN, we have
R(σ(1),σ(2))∈[−D2;D2]d×d.
Given the constant Kfrom (4.74), for any c>0, we define the set
Σ(2),k
N(U,ε):=n(σ(1),σ(2))∈ΣN(B(U,ε))2:kR(σ(1),σ(2))−Q(k)k2
F≥2K∆N(t)+co.
(4.80)
It is easy to see that by compactness we can find a finite covering of Σ(2),k
N(U,ε)by the
neighbourhoods (4.56) with centres, e.g., in the corresponding set of admissible overlap
matrices
V(k)
N(U,ε):=nR(σ(1),σ(1))∈[−D2;D2]d×d:(σ(1),σ(2))∈Σ(2),k
N(U,ε)o.
That is, there exists M=M(ε,δ)∈
N
and the finite collections of matrices {V(i)}M
i=1⊂
V(k)
N(U,ε)and {U(i)}M
i=1⊂B(U,ε)∩Sym+(d)such that
Σ(2),k
N(U,ε)⊂
M
[
i=1
Σ(2)
N(L∗(i),U(i),ε,δ),(4.81)
where
U(i):=U(i)V(i)
V∗(i)U(i)∈Sym+(2d),
and L∗(i)is the corresponding δ-minimal Lagrange multiplier.
4. Given i∈[1;M]∩
N
, let (n(i),x∗(i),Q∗(i),Λ∗(i)) be the corresponding toU(i)θ(i)-optimisers.
Due to Lipschitzianity of the Parisi functional (Proposition 4.2.2) and the fact that U(i)∈
B(U,ε)we can assume that n(i) = n. Using the bound (4.74) and the definition (4.80), we
obtain
ϕ(2)
N(k,t,x∗
i,Q∗
i,Σ(2)
N(L∗(i),U(i),ε,δ)) ≤2φ(x∗(i),Q∗(i),Λ∗(i))(t)−1
KkQ(k)−V(i)k2
F+C(ε+δ)
108 4 Estimates of Guerra’s remainder term in the SK model with multidimensional spins
≤2ϕN(t,x∗,Q∗,B(U,ε))−c+C(ε+δ),
where the last inequality is again due to Lipschitzianity of the Parisi functional (Propo-
sition 4.2.2) which allows to approximate functional’s value at (x∗(i),Q∗(i),Λ∗(i)) by the
value at (x∗,Q∗,Λ∗)paying the cost of at most Cε. Choose c>C(ε+δ). Then Lemma 4.3.2
implies that there exists L=L(ε,δ,c)>0 such that
µkΣ(2)
N(L∗,U,ε,δ)≤Lexp−N
L.
Therefore, the inclusion (4.81) gives
µkΣ(2),k
N(U,ε)≤LM exp−N
L.(4.82)
Hence, for each k∈[1;n]∩
N
, we have
µkhkRN(σ(1),σ(2))−Q(k)k2
Fi=µkhkRN(σ(1),σ(2))−Q(k)k2
F
1
Σ(2),k
N(U,ε)(σ(1),σ(2))i
+µkhkRN(σ(1),σ(2))−Q(k)k2
F1−
1
Σ(2),k
N(U,ε)(σ(1),σ(2))i
=: I +II.(4.83)
For all (σ(1),σ(2))∈ΣN(B(U,ε))2\Σ(2),k
N(U,ε,δ), we have by definition
kR(σ(1),σ(2))−Q(k)k2
F<2K∆N(t)+c.
Therefore, using Remark 4.1.4, we arrive to
II ≤2K∆N(t)+c.(4.84)
The bound (4.82) assures that
I≤LM exp−N
L.(4.85)
5. Combining (4.84) and (4.85) with (4.83) and (4.79), we obtain
d
dt∆N(t)≤2K∆N(t)+c+LM exp−N
L+C(ε+δ).
Hence,
d
dt(∆N(t)+c)exp(−2Kt)=exp(−2Kt)d
dt(∆N(t)+c)−2K(∆N(t) +c)
≤exp(−2Kt)d
dt(LM exp−N
L+C(ε+δ).
Integrating the above inequality and noting that due to (4.10) |∆N(0)|≤Cε, we arrive to
4.3 Remainder estimates 109
∆N(t)+c≤(Cε+c)exp(−2Kt)+ LM exp−N
L
+C(ε+δ)(exp(−2Kt)−1)+C(ε+δ).
Passing consequently to the limits N↑+∞,ε↓+0, δ↓+0 and finally c↓+0 in the above
inequality, we get
lim
ε↓+0∆(t)≤0,for all t∈[0;t0].
The existence of the N↑+∞limits is guaranteed by the general result of Guerra & Toninelli
(2003). The limits ε↓+0, δ↓+0 exist due to monotonicity. Finally, combining the above
inequality with (4.78), we get
lim
ε↓+0∆(t) = 0,for all t∈[0;t0]. (4.86)
6. Now it is easy to extend the validity of (4.86) onto the whole interval [0;1]. Indeed, due to
the boundedness of the derivatives of ϕNand φ, we have, for any t∈[0;1],
∆N(t)≤Z1
0
d
dt∆N(t)dt
=Zt0
0
+Z1
t0d
dt∆N(t)dt
≤∆N(t0)−∆N(0)+Z1
t0
d
dt∆N(t)dt
≤∆N(t0)+L(1−t0).(4.87)
Passing to the N↑+∞limit, applying (4.86), and then to t0→1 limit in (4.87), we get
lim
ε↓+0∆(t) = 0,for all t∈[0;1].
7. In particular, the previous formula yields
0=lim
ε↓+0∆(1) = φ(x∗,Q∗,Λ∗)(1)−lim
ε↓+0ϕN(1,x∗,Q∗,B(U,ε)).
Note that ϕN(1,x,Q,B(U,ε)) does not depend on the choice of xand Q. Hence, by (4.77),
we obtain
|lim
ε↓+0ϕN(1,x∗,Q∗,B(U,ε))−inf
ρ∈Q0(U,d)
Λ∈Sym(d)
P(β,ρ,U,Λ)|≤θ.
The proof of (4.76) is finished by noticing that the θcan be made arbitrary small.
ut
5
The SK model with multidimensional Gaussian spins
In this chapter, we prove the local Parisi formula (Theorem 2.3.10) for the SK model with
multidimensional Gaussian spins.
5.1 Introduction
Let Σ:=
R
dand fix some vector h∈
R
d. Let µ∈Mf(Σ)be the finite measure with the following
density (with respect to the Lebesgue measure λon Σ)
dµ
dλ(σ) = detC
(2π)d!1/2
exp−1
2hCσ,σi+hh,σi,(5.1)
where C∈Sym+(d). Note that, given m∈
R
dand C∈Sym+(d)such that detC6=0, the density
(5.1) with h:=Cm coincides (up to the constant factor exp−1
2hCm,mi) with the Gaussian
density with covariance matrix C−1and mean m.
Remark 5.1.1. It turns out that only matrices C with sufficiently large eigenvalues will result
in finite global free energy, cf. Lemma 5.2.8. The local free energy is, in contrast, always finite,
see Lemma 5.2.7 and Theorem 5.1.1.
Consider the function f:(0 : +∞)2→
R
given by
f(c,u) = (β2u2+logcu −cu +1,u∈(0; √2
2β],
(2√2β−c)u+log c
β−1
2(1+log2),u∈(√2
2β;+∞].(5.2)
The following result shows that, at least, in the highly symmetric situation (5.1) with h=0 the
multidimensional Parisi formula indeed holds true (see Lemma 5.2.7 for an explanation why
the result is indeed a Parisi-type formula).
Theorem 5.1.1. Let µsatisfy (5.1) with h =0. Assume that the matrices U and C are simultane-
ously diagonalisable in the same basis. Denote by {cv∈
R
+}d
v=1and {uv∈
R
+}d
v=1the eigenval-
ues of the matrices C and U, respectively. Moreover, assume that minvuv>0and minvcv>0.
Then we have
lim
ε↓+0lim
N↑+∞pN(ΣN(B(U,ε))) =
d
∑
v=1
f(cv,uv).
112 5 The SK model with multidimensional Gaussian spins
Remark 5.1.2. Close results have previously been obtained in the case of the spherical model by
Panchenko & Talagrand (2007b), from where we borrow the general methodology of the proof of
the above theorem. As noted by Panchenko & Talagrand (2007b), another more straightforward
way to obtain the Theorem 5.1.1 is to diagonalise the interaction matrix G and use the properties
of the corresponding random matrix ensemble.
5.2 Proof of the local Parisi formula
In this section, we prove Theorem 5.1.1. The rich symmetries of the Gaussian a priori distri-
bution allow rather explicit computations of the X0terms (see (3.10)). This allows us to prove
that the analogon of Assumption 4.3.1 is satisfied, implying the Parisi formula for the local free
energy (Theorem 5.1.1).
Remark 5.2.1. The case of Gaussian spins is very tractable due to the (unusually) good sym-
metry (i.e., the rotational invariance) of the Gaussian measure. Therefore, it is not surprising
that in this case the calculus resembles the one for the spherical SK model, cf. Panchenko &
Talagrand (2007b); Talagrand (2006a).
We start from the estimates under a generic (i.e., no simultaneous diagonalisation, cf. Sec-
tion 4.2.3) scenario.
5.2.1 The case of positive increments
Let, for k∈[0;n]∩
N
,
∆Q(k):=Q(k+1)−Q(k).
We define, for Λ∈Sym(d), a family of matrices nD(l)∈
R
d×don+1
l=0as follows
D(n+1):=C,
and, further, for k∈[0;n]∩
N
,
D(k):=C−Λ−2β2n
∑
l=k
xl∆Q(l).(5.3)
We assume that the matrices Λand Care such that, for all l∈[1;n+1]∩
N
, we have
D(l)0.
We need the following two small (and surely known) technical Lemmata which exploit the
symmetries of our Gaussian setting. We include their statements for reader’s convenience.
Lemma 5.2.1. Fix some vector h ∈
R
dand a Gaussian random vector z ∈
R
dwith Varz=C−1∈
R
d×d.
Then we have
E
z[exp(hz,hi+hΛσ,σi)] =dethC(C−Λ)−1i1/2
×exp1
2(C−Λ)−1h,h.
5.2 Proof of the local Parisi formula 113
Proof. This is a standard Gaussian averaging argument.
ut
Lemma 5.2.2. For a positive definite matrix ∆Q∈Sym(d), let z ∼N(0,∆Q). We fix also
another positive definite matrix D ∈Sym(d)such that ∆Q−1D−1.
Then we have
E
zexp1
2hD−1(z+h),z+hi=detD(D−∆Q)−1−1/2
×Z
R
dexp1
2h(D−∆Q)−1h,hi.
Proof. This is a standard Gaussian averaging argument. See, e.g., Talagrand (2006a) for an
argument in 1-D.
ut
Now we are ready to compute the term X0(x,Q,U,Λ)(see (3.10)) corresponding to the a
priori distribution (5.1) in a rather explicit way.
Lemma 5.2.3. We have
X0(x,Q,U,Λ) = 1
2 h[D(1)]−1,∆Q(0)i+h[D(1)]−1h,hi+
n
∑
l=1
1
xl
log detD(l+1)
detD(l)!!.
Proof. 1. We start from computing the following quantity
Xn+1:=logZ
R
dexp n
∑
l=0hY(l),σi+hΛσ,σi!dµ(σ),(5.4)
where Y(l)∈
R
dare independent Gaussian vectors with variance
VarhY(l)i=2β2∆Q(l).
We denote
eh:=h+
n
∑
l=0
Y(l).
Lemma 5.2.1 gives
Z
R
dexp n
∑
l=0hY(l),σi+hΛσ,σi!dµ(σ) =dethC(C−Λ)−1i1/2
×exp1
2D(C−Λ)−1eh,ehE.
114 5 The SK model with multidimensional Gaussian spins
2. Next, we define, for l∈[0;n]∩
N
, recursively the following quantities
Xl:=1
xl
log
E
YlexpxlXl+1.
Applying the Lemma 5.2.2 to (5.4) recursively, we obtain
X1:=1
2h[(D(1)]−1Y(0)+h,Y(0)+hi+1
2
n
∑
l=1
1
xl
log detD(l+1)
detD(l)!.(5.5)
Recall that we have
X0=lim
x→+0
1
xlog
E
Y0expxX1
=
E
Y0X1(5.6)
and note that
E
Y0hh[D(1)]−1(Y(0)+h),Y(0)+hii=2β2h[D(1)]−1,∆Q(0)i+h[D(1)]−1h,hi.(5.7)
Hence, combining (5.6) and (5.7) with (5.5), we obtain the theorem.
ut
5.2.2 Simultaneous diagonalisation scenario
In what follows, we employ the simultaneous diagonalisation scenario introduced in Sec-
tion 4.2.3. Suppose that, for l∈[0;n+1]∩
N
, and some matrix O∈O(d), we have
D(l):=O∗d(l)O,
where the vectors d(l)∈
R
d, for l∈[0;n]∩
N
, satisfy
0≺d(l)≺d(l+1).
That is, the vectors d(l)are (component-wise) increasingly ordered and non-negative.
Lemma 5.2.4. We have
X0(x,Q,U,Λ) = 1
2
d
∑
v=1 2β2q(1)
v+h2
v
d(1)
v
+
n
∑
l=1
1
xl
log d(l+1)
v
d(l)
v!!,(5.8)
β2
2
n
∑
k=1
xkkQ(k+1)k2
F−kQ(k)k2
F=β2
2
n
∑
k=1
xlkq(k+1)k2
2−kq(k)k2
2.(5.9)
Proof. This is a standard argument which relies on the standard invariance properties of the
determinant and the matrix trace.
ut
5.2 Proof of the local Parisi formula 115
Define the 1-D Parisi functional for the case (5.1) as
P(ρ,λ):=−λu+2β2q(1)+h2
d(1)+
n
∑
l=1
1
xl
log d(l+1)
d(l)!
−β2n
∑
l=1
xl[q(l+1)]2−[q(l)]2.(5.10)
Proposition 5.2.1. There exists C =C(Σ)>0such that, for all u ∈
R
d
+and all ε,δ>0, there
exists an δ-minimal Lagrange multiplier λ=λ(U,ε,δ)∈
R
din (3.11) such that, for all t ∈[0;1]
and all (x,ρ), we have
pN(ΣN(V(Λ,U,ε,δ))) ≤1
2inf
ρ,λ d
∑
v=1
P(ρv,λv)!+C(ε+δ)(5.11)
and
lim
N↑+∞pN(ΣN(B(U,ε))) ≥1
2inf
ρ,λ d
∑
v=1
P(ρv,λv)+ lim
N↑+∞Z1
0
R(t,x,Q,ΣN(B(U,ε)))dt!
+C(ε+δ).(5.12)
Proof. We combine (5.8) and (5.9) and the Proposition 4.1.1 to get (5.11) and (5.12). ut
5.2.3 The Crisanti-Sommers functional in 1-D
In this subsection, we adapt the proof of Talagrand (2006a) to obtain the equivalence between
the (very tractable) Crisanti-Sommers functional (Crisanti & Sommers, 1992) and the Parisi one
(5.10) in the case of the Gaussian a priori measure (5.1). Similar ideas based on the symmetry
of the a priori measure were exploited in the case of the spherical models by Ben Arous et al.
(2001); Panchenko & Talagrand (2007b).
We restrict the consideration to 1-D situation for a moment. Given u≥0, consider ρ∈
Q0
n(u,1),λ∈
R
,h∈
R
and let {d(l)∈
R
}n+1
l=1be the scalars playing the role of matrices D(l)(cf.
(5.3)). That is,
d(l):=c−λ−2β2n
∑
k=l
xkq(k+1)−q(k),
d(n+1):=c.
We define, for k∈[1;n]∩
N
, the family of vectors {s(k)∈
R
d}n
k=0by
s(k):=
n
∑
l=k
xlq(l+1)−q(l).(5.13)
We also define the Crisanti-Sommers functional as follows
C S (ρ):=1−cu +h2s(1)+q(1)
s(1)+
n−1
∑
l=1
1
xl
log s(l)
s(l+1)!+loghc(u−q(n))i
+β2n
∑
l=1
xl[q(l+1)]2−[q(l)]2.(5.14)
116 5 The SK model with multidimensional Gaussian spins
Lemma 5.2.5. If (ρ,λ)is an optimiser for (5.10), that is,
P(ρ,λ) = inf
(ρ0,λ0)
P(ρ0,λ0),(5.15)
then, for all k ∈[1;n]∩
N
, the pair (ρ,λ)satisfies
q(k)=h2+2β2q(1)
[d(1)]2+
k−1
∑
l=1
1
xl1
d(l)−1
d(l+1).(5.16)
Moreover,
λ=c−2β2(u−q(n))−(u−q(n))−1,(5.17)
and, for all k ∈[1;n]∩
N
, we have
1
s(k+1)−1
s(k)=2β2xkq(k+1)−q(k),(5.18)
and also
s(k)=1
d(k).(5.19)
Remark 5.2.2. In the formulation of the theorem (as well as elsewhere), it is implicit that
d(k)=d(k)(ρ,λ)and s(k)=s(k)(ρ,λ).
Proof. 1. Rearranging the terms in (5.10), we observe that
P(ρ0,λ0) =−λu+2β2q(1)+h2
d(1)+
n
∑
l=2
logd(l)1
xl−1−1
xl+1
xn
logd(n+1)−1
x1
logd(1)
−β2n
∑
l=1
xl[q(l+1)]2−[q(l)]2.(5.20)
We compute, for k,l∈[1;n]∩
N
,
∂d(l)
∂q(k)=
0,k<l,
2β2xk,l=k,
2β2xk−xk−1,k>l.
(5.21)
Using (5.21) and the representation (5.20), we compute the necessary condition for (q,λ)
satisfy (4.72), for k∈[2;n]∩
N
,
0=∂
∂q(k)P(q,λ) =2β2xk−xk−1"−2β2q(1)+h2
[d(1)]2+
k−1
∑
l=2
1
d(l)1
xl−1−1
xl
+1
d(k)xk−1−1
x1d(1)+qk#.(5.22)
5.2 Proof of the local Parisi formula 117
We also have (for k=1)
0=∂
∂q(1)P(q,λ) = 2β2
d(1)−x1q(1)+h2
[d(1)]2−x1
x1d(1)+x1q(1)
=2β2x1
q(1)−q(1)+h2
[d(1)]2
.(5.23)
Relations (5.22) and (5.23) then imply (5.16).
2. Using the fact that
∂d(l)
∂λ =−1,
we obtain
∂
∂λ P(q,λ) = −u+h2+2β2q(1)
[d(1)]2+
n−1
∑
l=1
1
xl1
d(l)−1
d(l+1)+1
d(n).(5.24)
Applying (5.16) with k=nin (5.24), we obtain that the necessary condition for λto satisfy
(5.15) is as follows
0=∂
∂λ P(q,λ) = −u+q(n)+1
xn1
d(n)−1
d(n+1)
=−u+q(n)+1
d(n)=−u+q(n)+c−λ−2β2(u−q(n))−1
(5.25)
which implies (5.17).
3. Relation (5.18) is proved as follows. Subtracting the relations (5.16), we obtain, for k∈
[1;n−1]∩
N
,
xkq(k+1)−q(k)=1
d(k)−1
d(k+1).(5.26)
By (5.25), we have
xnq(n+1)−q(n)=u−q(n)=1
d(n).
(That is, (5.26) is valid also for k=n.) Combining the previous two relations, we get, for
k∈[1;n]∩
N
,
s(k)=1
d(k).(5.27)
Using (5.27) and (5.26), we get
2β2xkq(k+1)−q(k)=d(k+1)−d(k)
118 5 The SK model with multidimensional Gaussian spins
(by (5.26)) =d(k+1)d(k)xkq(k+1)−q(k)=d(k+1)d(k)s(k)−s(k+1)
(by (5.27)) =1
s(l+1)−1
s(l)
which is (5.18).
ut
Lemma 5.2.6. If ρis an optimiser of (5.14), that is,
C S (ρ) = inf
ρ0
C S (ρ0),
then, for all l ∈[1;n]∩
N
,(5.18) holds.
Proof. The strategy is the same as in the previous lemma. We rearrange the summands in (5.14)
to get
C S (ρ) =h2s(1)+q(1)
s(1)+logs(1)
x1−logs(n)
xn−1
+
n−1
∑
l=21
xl−1
xl+1logs(l)
+logc(u−q(n))+β2n
∑
l=1
xl[q(l+1)]2−[q(l)]2.(5.28)
We have, for k,l∈[1;n]∩
N
,
∂s(l)
∂q(k)=
0,k<l,
−xk,k=l,
xk−1−xk,k>l.
(5.29)
1. Relation (5.29) implies, for k∈[2;n−1]∩
N
,
∂
∂q(k)C S (ρ) =h2(xk−1−xk)−q(1)
[s(1)]2(xk−1−xk)+ xk−1−xk
x1s(1)
+
k−1
∑
l=2
xk−1−xk
s(l)1
xl−1
xl−1−xk
s(k)1
xk−1
xk−1
+2β2q(k)xk−1−xk=0.
Hence,
2β2q(k)=−h2+q(1)
[s(1)]2−1
x1s(1)+1
xk−1s(k)−
k−1
∑
l=2
1
s(l)1
xl−1
xl−1
=−h2+q(1)
[s(1)]2−
k−1
∑
l=1
1
xl1
s(l)−1
s(l+1).(5.30)
2. To handle the case k=n, we note that
log1+c(u−q(n))=1
xn
log s(n)
s(n+1)!,
and, hence, the argument in the previous item shows that (5.30) is also valid for k=n.
5.2 Proof of the local Parisi formula 119
3. Differentiating the representation (5.28) with respect to q(1)and using (5.29), we obtain
∂
∂q(1)C S (ρ) = −x1h2+1
s(1)+x1q(1)
[s(1)]2−x1
x1s(1)−2β2x1q(1)=0.
Therefore,
2β2q(1)=−h2+q(1)
[s(1)]2
which is (5.30), for k=1.
4. Subtracting equations (5.30), we arrive to (5.18), for all k∈[1;n]∩
N
.
ut
Proposition 5.2.2. The functionals (5.14) and (5.10) are equivalent in the following sense
inf
ρ0,λ0
P(ρ0,λ0) = inf
ρ0
C S (ρ0).
Proof. 1. Let (ρ,λ)be the solutions of equations (5.18) and (5.17). Lemma 5.2.6 guarantees
that ρis the optimiser of the Crisnati-Sommers functional and Lemma 5.2.5 assures that
(ρ,λ)is the optimiser of the Parisi functional.
2. We have
P(ρ,λ)−C S (ρ) =−λu+2β2q(1)s(1)−q(1)
s(1)+cu −1
−2β2n
∑
l=1
xl[q(l+1)]2−[q(l)]2.(5.31)
We can simplify the Φ[B]-like term (that is the summation) in (5.31), using (5.18) and (5.17).
Indeed,
2β2n−1
∑
l=1
xl[q(l+1)]2−[q(l)]2=2β2n−1
∑
l=1
xlq(l+1)[q(l+1)−q(l)]+ q(l)[q(l+1)−q(l)]
(by (5.18) and (5.13)) =
n−1
∑
l=12β2q(l+1)hs(l)−s(l+1)i+q(l)1
s(l+1)−1
s(l).
(5.32)
Regrouping the summands in (5.32), we get
(5.32) =2β2n−1
∑
l=1
s(l)q(l+1)−q(l)+2β2q(1)s(1)−q(n)s(n)
+
n−1
∑
l=1
q(l)−q(l+1)
s(l+1)+ q(n)
s(n)−q(1)
s(1)!.(5.33)
Due to (5.18), we have
120 5 The SK model with multidimensional Gaussian spins
2β2q(l+1)−q(l)=s(l)−s(l+1)
xls(l)s(l+1)=q(l+1)−q(l)
s(l)s(l+1).
Applying the previous relation, we get that the both summations in (5.33) cancel out and we
end up with
(5.33) =2β2q(1)s(1)−q(n)s(n)+q(n)
s(n)−q(1)
s(1).
Now, turning back to (5.31), we get
P(ρ,λ)−C S (ρ) = −λu−2β2u2−[q(n)]2+2β2q(n)s(n)−q(n)
s(n)+cu −1
(by (5.17)) and (5.13) =−uc−2β2(u−q(n))−(u−q(n))−1−2β2u2−[q(n)]2
−q(n)
u−q(n)+2β2q(n)u−q(n)+cu −1
=0.
ut
5.2.4 Replica symmetric calculations
In this subsection, we shall consider the one dimensional case of the a priori measure (5.1) with
h=0. We shall also restrict the computations to the case n=1 which is often referred to in
physical literature as the replica symmetric scenario. It is indeed the right scenario under the
above assumptions, as shows Theorem 5.1.1.
Lemma 5.2.7. Let µsatisfy (5.1) with h =0. Assume d =1, n =1and c >0. Given u ≥0, we
have
inf
ρ∈Q(u,1)
C S (ρ) = inf
q∈[0;u]1−cu +log(c(u−q))+ q
u−q+β2u2−q2=f(c,u),(5.34)
where f (c,u)is defined in (5.2).
Proof. Using the definitions, we obtain
∂
∂qC S (ρ) = ∂
∂qlog(u−q)+ q
u−q+β2u2−q2=q
(u−q)2−2β2q.
Hence, the critical points of q7→C S (q,u)are
q0=0,q1,2=u±√2
2β.
Furthermore, we also have
∂2
∂q2C S (q,u) = 1
(u−q)2+2q
(u−q)3−2β2.
5.2 Proof of the local Parisi formula 121
Hence, as a simple calculation shows, the infima in (5.34) are attained on
q∗=(0,u≤√2
2β,
u−√2
2β,u>√2
2β
(5.35)
which implies (5.34). ut
Lemma 5.2.8. Under the assumptions of Lemma 5.2.7, we have
1. For c ≥2√2β, we have
sup
u≥0
inf
q∈[0;u]
C S (q,u) = C S (0,u∗) = β2(u∗)2+logcu∗−cu∗+1,
where
u∗:=1
4β2c−pc2−8β2.
2. For c <2√2β, we have
sup
u≥0
inf
q∈[0;u]
C S (q,u)=+∞.
Remark 5.2.3. Under the assumptions, the above theorem says that from the point of view of
the global free energy, the system can only exist in the “high temperature” scenario, cf. (5.2).
The threshold at c0=2√2βcould be easily understood from the perspective of the norms of
random matrices.
Proof. 1. Suppose c≥2√2β. Recalling (5.2), for u∈(0; √2
2β], we introduce the following func-
tion
f(u):=log(cu)+β2u2−cu +1.
We have
∂
∂uf(u) = 1
u+2β2u−c.
Hence, the critical points of the function fare
u1,2=c±pc2−8β2
4β2.
Furthermore, we have
∂2
∂u2f(u) = 2β2−1
u2.
We notice that u∗≤√2
2βand, hence, due to (5.2)
C S (0,u∗) = β2(u∗)2+logcu∗−cu∗+1.
2. If c<2√2β, then the function
u7→(2√2β−c)u+log c
β−1
2(1+log2)
is unbounded on (√2
2β;+∞).
ut
122 5 The SK model with multidimensional Gaussian spins
5.2.5 The multidimensional Crisanti-Sommers functional
Recall the definition (4.73).
Proposition 5.2.3. Assume d =1. Given u >0, we have
2φ(x∗,Q∗,Λ∗)(t) =
3√2β−cu+log c
β−1−log2
2−t√2uβ−1
2,u>√2
2β,
2β2(u)2+log(cu)−cu +1−tβ2(u)2,u≤√2
2β.(5.36)
Proof. Combining (5.10), (5.14) with Lemma 5.2.7 and Proposition 5.2.2, we get the claim.
ut
5.2.6 Talagrand’s a priori estimates
In this subsection, we prove that Assumption 4.3.1 is satisfied in the case of the Gaussian a
priori distribution (5.1) with h=0.
Theorem 5.2.1. Let µsatisfy (5.1) with h =0, assume U ∈Sym+(d)is such that minvuv>√2
2β
and suppose C 0. Let Q =Q∗and Λ=Λ∗.
Then, for any t0∈(0;1)and any t ∈(0;t0], we have (cf. (4.74) with k =1)
ϕ(2)
N(1,t,x,Q,Σ(2)
N(L,U,ε,δ)) ≤2φ(x,Q,Λ)(t)−1
KkQ(1)−Vk2
F+O(ε+δ).(5.37)
Proof. 1. We employ the notations of Section 4.3.2. Let n=1. Given U∈Sym(2d)(cf. (4.51)),
choose arbitrary matrices nQ(l)∈Sym(2d)|l∈[0;2]∩
N
osatisfying (4.52). Define x:=x
which, in particular, implies that ζ=ξ. Finally, we set, for l∈[0;n+1]∩
N
,e
Q(l):=Q(l).
2. Proposition 4.3.1 implies that, for any δ-minimal L∈
R
2d×2d, we have
ϕ(2)
N(1,t,x,Q,Σ(2)
N(L,U,ε,δ)) ≤−hL,Ui−tβ2
2kQ(2)k2
F−kQ(1)k2
F
+X(2)
0(1,x,b
Q(l)(t),L)+O(ε+δ).(5.38)
3. We define a matrix C∈
R
2d×2das follows
C:=C0
0C.
Recalling (5.3), we define also the following matrices D(2):=Cand
D(1):=C−L−b
Q(2)(t)−b
Q(1)(t).(5.39)
Applying the Proposition 5.2.1 to (5.38), we get
ϕ(2)
N(1,t,Σ(2)
N(L,U,ε,δ)) ≤1
2h−hL,Ui−tβ2kQ(2)k2
F−kQ(1)k2
F
+2β2h[D(1)]−1,b
Q(1)(t)i+log detD(2)
detD(1)!#+O(ε)
=:e
Φ(2),k,x,L+O(ε).(5.40)
5.2 Proof of the local Parisi formula 123
4. Assume that the matrices
Q(1),Q(2),D(1)∈
R
2d×2d(5.41)
are simultaneously diagonalisable in the same basis which is given by the orthogonal matrix
O∈
R
2d×2d. Let the vectors
q(1),q(2),d(1)∈
R
2d(5.42)
be the corresponding spectra of the matrices (5.41). That is, we assume that
Q(1)=O∗diagq(1)O,Q(2)=O∗diagq(2)O,
D(1)=O∗d(1)O,e
Q(1)=O∗diag e
Q0(1)O,
where we have introduced the matrix e
Q0(1)(t)∈Sym+(2d). By (5.35), we have, Q(2)−
Q(1)=√2
2βI, where Idenotes the unit matrix of the suitable dimension. The definitions (4.62)
and (4.63) then imply
e
Q(2)−e
Q(1)=√2
2βI.(5.43)
Using the definitions and the above relation, we obtain
b
Q(1)
v(t) = O∗tdiagq(1)+(1−t)e
Q0(1)O,
b
Q(2)(t)−b
Q(1)(t) = O∗tdiag(q(2)−q(1))+(1−t)√2
2βIO.(5.44)
Motivated by (5.19), we set
d(1)
v:=uv−q(1)
v−1.(5.45)
In view of (5.39), the above choice necessarily yields (cf. (5.17))
L=C−O∗diag(uv−q(1)
v)−1O−b
Q(2)(t)−b
Q(1)(t)
=C−O∗diag(uv−q(1)
v)−1+tdiag(q(2)−q(1))+(1−t)√2
2βIO.(5.46)
Applying Lemma 5.2.4 to (5.40) and using (5.46), (5.45), (5.44), we get the following diag-
onalised representation of (5.38)
ϕ(2)
N(1,t,x,Q,Σ(2)
N(L,U,ε,δ)) ≤1
2logdetC−1
2hC,Ui
+1
2
2d
∑
v=1nuvh(uv−q(1)
v)−1+2β2t(q(2)
v−q(1)
v)+(1−t)√2
2βi
+2β2(uv−q(1)
v)tq(1)
v+(1−t)eq(1)
v+log(uv−e
Q0(1)
v,v)
124 5 The SK model with multidimensional Gaussian spins
−tβ2(q(2)
v)2−(q(1)
v)2o+O(ε).(5.47)
Using the definitions, we get
hC,Ui=2hC,Ui=2
d
∑
v=1
cvuv,
logdetC=2logdetC=2
d
∑
v=1
logcv.(5.48)
Motivated by (5.43) (or by (5.35)), we define
q(1)
v:=uv−√2
2β.(5.49)
In this case, as a straightforward calculation shows, the expression in the curly brackets in
(5.47) equals
2√2βuv+β√2e
Q0(1)
v,v(1−t)−logβ−1
2(log2 −t).(5.50)
By the definitions and the general properties of matrix trace, we have
2d
∑
v=1e
Q0(1)
v,v=
2d
∑
v=1e
Q(1)
v,v=2
d
∑
v=1
Q(1)
v,v,
2d
∑
v=1
uv=2
d
∑
v=1
Uv,v.(5.51)
Combining (5.47) with (5.50), (5.51) and (5.48), we obtain
ϕ(2)
N(1,t,x,Q,Σ(2)
N(L,U,ε,δ)) ≤
d
∑
v=1−cvuv+logcv+3√2uβ
−1
2(log2 −t)−√2βtu −logβ−1+O(ε)
=2
d
∑
v=1
φ(t)|c=cv,
u=uv
+O(ε),(5.52)
where in the last line we have used the relation (5.36).
5. To get the version of the a priori bound (5.52) with the quadratic correction term as stated in
(5.37), we perturb the r.h.s of (5.38) around our choice of D(1)in (5.45), i.e.,
D(1)=Uv−Q(1)
v−1=√2βI,
where in the last equality we used (5.49).
ut
5.2.7 The local low temperature Parisi formula
Proof of Theorem 5.1.1. The result follows from Theorem 5.2.1 and Theorem 4.3.1. Note that
the proof of Theorem 4.3.1 requires a minor modification to cope with the fact that the a priori
distribution (5.1) is unbounded. This minor problem can be fixed by considering the pruned
Gaussian distribution and using the elementary estimates to bound the tiny Gaussian tails. ut
6
The GREM in the presence of uniform external field
In this chapter, we find the fluctuations of the ground state and of the partition function for the
GREM with external field. We provide an explicit expression for the free energy of the model.
We also obtain some large deviation results providing an expression for the free energy for a
class of models with Gaussian Hamiltonians and external field.
6.1 Introduction
Despite the recent substantial progress due to Guerra (2003), Aizenman et al. (2003, 2007),
and Talagrand (2006b) in establishing rigorously the Parisi formula for the free energy of the
celebrated SK model, understanding of the corresponding limiting Gibbs measure is still very
limited.
Due to the above mentioned works, it is now rigorously known that Derrida’s GREM is
closely related to the SK model at the level of free energy, see, e.g., (Bovier, 2006, Section 11.3).
Recently Bovier & Kurkova (2003a, 2004a,b, 2007) have performed a detailed study of the
geometry of the Gibbs measure for the GREM. This confirmed the predicted in the theoretical
physics literature hierarchical decomposition of the Gibbs measure in rigorous terms.
As pointed out in Bovier & Kurkova (2004a) (see also Ben Arous et al. (2005)), the GREM-
like models may represent an independent interest in various applied contexts, where correlated
heavy-tailed inputs play an important role, e.g., in risk modelling.
One of the key steps in the results of Bovier & Kurkova (2004a) is the identification of the
fluctuations of the GREM partition function in the thermodynamic limit with Ruelle’s proba-
bility cascades. In this chapter we also perform this step and study the effect of external field
on the fluctuations (i.e., the weak limit laws) of the partition function of the GREM in the ther-
modynamic limit. We find that the main difference introduced by the presence of external field,
comparing to the system without external field, is that the coarse graining mechanism should be
altered. This change reflects the fact that the coarse-grained parts of the system tend to have a
certain optimal magnetisation as prescribed by the strength of external field and by parameters
of the GREM. We use the general line of reasoning suggested by Bovier & Kurkova (2004a),
i.e., we consider the point processes generated by the scaling limits of the GREM Hamiltonian.
We streamline the proof of the weak convergence of these point processes to the corresponding
Poisson point process by using the Laplace transform.
126 6 The GREM in the presence of uniform external field
Organisation of the chapter
In the following subsections of the introduction we define the model of interest and formulate
our main results on the fluctuations of the partition function of the REM and GREM with ex-
ternal field and also on their limiting free energy (Theorems 6.1.1, 2.3.1, 6.1.3 and 6.1.4). Their
proofs are provided in the subsequent sections. Section 6.2 is devoted to the large deviation
results providing an expression for the free energy for a class of models with Gaussian Hamil-
tonians and external field (Theorem 6.2.1). In Section 6.3 we resort to more refined analysis and
perform the calculations of the fluctuations of the ground state and of the partition function in
the REM with external field in the thermodynamic limit. Section 6.4 contains the proofs of the
results on the fluctuations of the ground state and of the partition function for the GREM with
external field.
Definition of the model
In contrast to the work of Derrida & Gardner (1986), we consider here the GREM with ex-
ternal field which depends linearly on the total magnetisation (i.e., we consider the uniform
magnetic field, cf. the definitions (0.6) and (0.7)). Derrida & Gardner (1986) considered the
“lexicographic” external field which is particularly well adapted to the natural lexicographic
distance generated by the GREM Hamiltonian.
The important quantities are the free energy defined as
pN(β,h):=1
NlogZN(β,h),(6.1)
and the ground state energy
MN(h):=N−1/2max
σ∈ΣN
XN(h,σ).(6.2)
In what follows, we shall think of βand has fixed parameters. We shall occasionally lighten
our notation by not indicating the dependence on these parameters explicitly.
We denote the total magnetisation (cf. the second summand in (0.6)) by
mN(σ):=1
N
N
∑
i=1
σi,σ∈ΣN.(6.3)
In this chapter, we shall mainly be interested in the weak limit theorems (i.e., fluctuations) of
the partition function (0.7) and of the ground state as N↑+∞. To be precise, the general results
on Gaussian concentration of measure imply that (6.2) and (0.10) are self-averaging. By the
fluctuations of the ground state, we mean the weak limiting behaviour of the rescaled point
process generated by the Gaussian process (0.6). This behaviour is studied in Theorems 6.1.1
and 6.1.2 below. These theorems readily imply the formulae for the limiting free energy (0.10)
and the ground state (6.2). A recent account of mathematical results on the GREM without
external field and, in particular, on the behaviour of the limiting Gibbs measure can be found in
Bovier & Kurkova (2007). The GREM with external field was previously considered by Jana &
Rao (2006) (see also Jana (2007)), where its free energy was expressed in terms of a variational
problem induced by an application of Varadhan’s lemma. In this chapter, we apply very different
methods to obtain precise control of the fluctuations of the partition function for the GREM with
6.1 Introduction 127
external field. As a simple consequence of these results, we also get a rather explicit1formula
for the limiting free energy in the GREM with external field (see Theorem 6.1.4).
Main results
Limiting objects
We now collect the objects which appear in weak limit theorems for the GREM partition func-
tion and for the ground states. We denote by I:[−1;1]→
R
+Cram´
er’s entropy function, i.e.,
I(t):=1
2[(1−t)log(1−t)+(1+t)log(1+t)].(6.4)
Define
µ(t):=p2(log2 −I(t)),
M(h):=max
t∈[−1;1](µ(t)+ht).(6.5)
Suppose that the maximum in (6.5) is attained at t=t∗=t∗(h). (The maximum exists and is
unique, since µ(t)+ ht is strictly concave.) Consider the following two real sequences
AN(h):=µ(t∗)√N−1,(6.6)
BN(h):=M(h)√N+AN(h)
2logAN(h)2(I00(t∗)+ h)
2π(1−t2
∗).(6.7)
Define the REM scaling function uN,h(x):
R
→
R
as
uN,h(x):=AN(h)x+BN(h).(6.8)
Given f:D⊂
R
→
R
+, we denote by PPP(f(x)dx,x∈D)the Poisson point process with inten-
sity f. We start from a basic limiting object. Assume that the point process P(1)on
R
satisfies
P(1)∼PPP(exp(−x)dx,x∈
R
),(6.9)
and is independent of all random variables around. The point process (6.9) is the limiting object
which appears in the REM.
Theorem 6.1.1 (fluctuations of the ground state of the REM with external field).If n =1(the
REM case), then, using the above notations, we have
∑
σ∈ΣN
δu−1
N,h(XN(h,σ))
w
−−−→
N→∞
P(1),(6.10)
where the convergence is the weak one of the random probability measures equipped with the
vague topology.
1In contrast to (Jana & Rao, 2006, Theorem 5.1) and (Jana, 2007, Corollary 4.3.5), who stop at the level of variational
problem.
128 6 The GREM in the presence of uniform external field
To formulate the weak limit theorems for the GREM (i.e., for the case n>1), we need
a limiting object which is a point process closely related to the Ruelle probability cascade,
(Ruelle, 1987). Define, for j,k∈[1;n+1]∩
N
,j<k, the “slopes” corresponding to the function
ρin (0.5) as
θj,k:=qk−qj−1
xk−xj−1
.
Define also the following h-dependent “modified slopes”
e
θj,k(h):=θj,kµ(t∗(θ−1/2
j,kh))−2.
Define the increasing sequence of indices {Jl(h)}m(h)
l=0⊂[0;n+1]∩
N
by the following algo-
rithm. Start from J0(h):=0, and define iteratively
Jl(h):=minnJ∈[Jl−1;n+1]∩
N
:e
θJl−1,J(h)>e
θJ+1,k(h),for all k>Jo.(6.11)
Note that m(h)≤n. The subsequence of indices (6.11) induces the following coarse-graining
of the initial GREM
¯ql(h):=qJl(h)−qJl−1(h),(6.12)
¯xl(h):=xJl(h)−xJl−1(h),(6.13)
¯
θl(h):=θJl−1,Jl.(6.14)
The parameters (6.12) induce the new order parameter ρ(J(h)) ∈Q0
min the usual way
ρ(J(h))(q):=
m(h)
∑
l=1
qJl(h)
1
[xJl(h);xJl+1(h))(x).
Define the GREM scaling function uN,ρ,h:
R
→
R
as
uN,ρ,h(x):=
m(h)
∑
l=1h¯ql(h)1/2B¯xl(h)N¯
θl(h)−1/2hi+N−1/2x.
Define the rescaled GREM process as
GREMN(h,σ):=u−1
N,ρ,h(GREMN(h,σ)).
Define the point process of the rescaled GREM energies ENas
EN(h):=∑
σ∈ΣN
δGREMN(h,σ).(6.15)
Consider the following collection of independent point processes (which are also independent
of all random objects introduced above)
6.1 Introduction 129
{P(k)
α1,...,αl−1|α1,...,αl−1∈
N
;l∈[1;m]∩
N
}
such that
P(k)
α1,...,αk−1∼P(1).
Define the limiting GREM cascade point process Pmon
R
mas follows
Pm:=∑
α∈
N
m
δ(P(1)(α1),P(2)
α1(α2),...,P(m)
α1,α2,...,αm−1(αm)),(6.16)
Consider the following constants
¯
γl(h):=e
θJl−1,Jl1/2,
and define the function Eh,f:
R
m→
R
as
E(m)
h,ρ(e1,...,em):=¯
γ1(h)e1+...+¯
γm(h)em.
Note that due to (6.11), the constants {¯
γl(h)}m
l=1form a decreasing sequence, i.e., for all l∈
[1;m]∩
N
, we have
¯
γl(h)>¯
γl+1(h).(6.17)
The cascade point process (6.16) is the limiting object which describes the fluctuations of the
ground state in the GREM.
Theorem 6.1.2 (fluctuations of the ground state of the GREM with external field).We have
EN(h)w
−−−→
N↑+∞Z
R
mδE(m)
h,ρ(e1,...,em)Pm(de1,...,dem)(6.18)
and
MN(h)−−−→
N↑+∞
m(h)
∑
l=1h¯ql(h)¯xl(h)1/2M¯
θl(h)−1/2hi,(6.19)
almost surely and in L1.
Theorem 6.1.2 allows for complete characterisation of the limiting distribution of the GREM
partition function. To formulate the result, we need the β-dependent threshold l(β,h)∈[0;m]∩
N
such that above it all coarse-grained levels l>l(β,h)of the limiting GREM are in the “high
temperature regime”. Below this threshold the levels l≤l(β,h)are in the “frozen state”. Given
β∈
R
+, define
l(β,h):=max{l∈[1;n]∩
N
:β¯
γl(h)>1}.
We set l(β,h):=0, if β¯
γ1(h)≤1. The following gives full information about the limiting
fluctuations of the partition function at all temperatures.
130 6 The GREM in the presence of uniform external field
Theorem 6.1.3 (fluctuations of the partition function of the GREM with external field).We
have
exp"−β√N
l(β,h)
∑
l=1¯ql(h)1/2B¯xl(h)N¯
θ−1/2
lh#
×exp−Nlog2 +logchβh(1−xJl(β,h))+1
2β21−qJl(β,h)ch2/3βh(1−xJl(β,h))
×ZN(β,h)w
−−−→
N↑+∞K(β,h,ρ)Z
R
l(β,h)exphβE(l(β,h))
h,ρ(e1,...,el(β,h))iPl(β,h)(de1,...,del(β,h)),
(6.20)
where the constant K(β,h,ρ)depends on β, h and ρonly. Moreover, K(β,h,ρ) = 1, if
βγl(β,h)+1<1and K(β,h,ρ)∈(0;1), if βγl(β,h)+1=1.
The above theorem suggests that the increasing sequence of the constants {βl:=¯
γ−1}m(h)
l=1⊂
R
+can be thought of as the sequence of the inverse temperatures at which the phase transitions
occur: at βlthe corresponding coarse-grained level lof the GREM with external field “freezes”.
As a simple consequence of the fluctuation results of Theorem 6.1.3, we obtain the following
formula for the limiting free energy of the GREM.
Theorem 6.1.4 (free energy of the GREM with external field).We have
lim
N↑+∞pN(β,h) =β
l(β,h)
∑
l=1h(¯xl¯ql)1/2µ(t∗(¯
θ−1/2
lh))+h¯xlt∗(¯
θ−1/2
lh)i
+log2 +logchβh(1−xJl(β,h))+1
2β21−qJl(β,h),(6.21)
almost surely and in L1.
6.2 Partial partition functions, external fields and overlaps
In this section, we propose a way to compute the free energy of disordered spin systems with
external field using the restricted free energies of systems without external field. The compu-
tation involves a large deviations principle. For gauge invariant systems, we also show that the
partition function of the system with external field induced by the total magnetisation has the
same distribution as the one induced by the overlap with fixed but arbitrary configuration. This
section is based on the ideas of Derrida & Gardner (1986).
Fix p∈
N
. Given some finite interaction p-hypergraph (VN,E(p)
N), where VN= [1;n]∩
N
and
E(p)
N⊂(VN)p, define the p-spin interaction Hamiltonian as
XN(σ):=∑
i∈E(p)
N
J(N,p)
iσi1σi2···σip,σ∈ΣN,(6.22)
where J(N,p):=nJ(N,p)
ioi∈E(p)
N
is the collection of random variables having the symmetric joint
distribution. That is, we assume that, for any ε(1),ε(2)∈{−1;+1}E(p)
N, and any t∈
R
E(p)
N,
6.2 Partial partition functions, external fields and overlaps 131
E
exp
i∑
r∈E(p)
N
trε(1)
rJ(N,p)
i
=
E
exp
i∑
r∈E(p)
N
trε(2)
rJ(N,p)
i
,(6.23)
where i ∈
C
denotes the imaginary unit.
A particular important example of (6.22) is Derrida’s p-spin Hamiltonian given by
SK(p)
N(σ):=N−p/2N
∑
i1,...,ip=1
gi1,...,ipσi1σi2···σip,
where {gi1,...,ip}N
i1,...,ip=1is a collection of i.i.d. standard Gaussian random variables. Note that
the condition (6.23) is obviously satisfied.
Given µ∈ΣN, define the corresponding gauge transformation Tµ:ΣN→ΣNas
Tµ(σ)i=µiσi,σ∈ΣN.(6.24)
Note that the gauge transformation (6.24) is obviously an involution. We say that a d-variate
random function f:Σd
N→
R
is gauge invariant, if, for any µ∈ΣNand any (σ(1),...,σ(d))∈Σd
N,
f(Tµ(σ(1)),...,Tµ(σ(d))) ∼f(σ(1),...,σ(d)),
where ∼denotes equality in distribution. Define the overlap between the configurations σ,σ0∈
ΣNas
RN(σ,σ0):=1
N
N
∑
i=1
σiσ0
i.(6.25)
Note that the overlap (6.25) and the lexicographic overlap (0.2) are gauge invariant.
Given a bounded function FN:ΣN→
R
, define the partial partition function as
Z(p)
N(β,q,ε,XN,FN):=∑
σ:|FN(σ)−q|≤ε
expβ√NXN(σ).(6.26)
Denote
UN:=FN(ΣN),U:=∞
[
N=1
UN.(6.27)
(The bar in (6.27) denotes closure in the Euclidean topology.) Note that for the case FN=RN
we obviously have
UN=1−2k
N:k∈[0;N]∩
Z
,U= [−1;1].
Proposition 6.2.1 (Derrida & Gardner (1986)).Assume that XNis given either by (6.22) or
XN∼GREMN. Fix some gauge invariant bivariate function FN:Σ2
N→
R
, and q ∈
R
.
Then, for all σ0,τ0∈ΣN, we have
132 6 The GREM in the presence of uniform external field
Z(p)
N(β,q,ε,XN,FN(·,σ0)) ∼Z(p)
N(β,q,ε,XN,FN(·,τ0)).(6.28)
In particular, the partial partition function (6.26) with FN:=RN(·,σ0)has the same distribution
as the partial partition function which corresponds to fixing the total magnetisation (6.3), i.e.,
Z(p)
N(β,q,ε,XN,RN(·,σ0)) ∼Z(p)
N(β,m,ε,):=∑
σ:|m(σ)−q|<ε
expβ√NXN(σ).
Remark 6.2.1. The proposition obviously remains valid for the Hamiltonians XNgiven by the
linear combinations of the p-spin Hamiltonians (6.22) with varying p ∈
N
.
Proof. 1. If XNis defined by (6.22), then (6.28) follows due to the gauge invariance of (6.22)
and FN. Indeed, there exists µ∈ΣNsuch that σ0=Tµ(τ0). Define
J(N,p,µ)
i:=J(N,p)
iµi1···µip.
Due to the symmetry of the joint distribution of J(N,p), we have
{XN(σ)}σ∈ΣN∼{XN(σ)|J(N,p)=J(N,p,µ)}σ∈ΣN
which implies (6.28).
2. If XN=GREMN, then, since XNis a Gaussian process, to prove the equality in distribution,
it is enough to check that the covariance of XNis gauge symmetric. Equivalence (6.28)
follows, due to (0.5) and the fact that the lexicographic overlap (0.2) is gauge invariant.
ut
The partial partition function (6.26) induces the restricted free energy in the usual way:
p(p)
N(β,q,ε,XN,FN):=1
NlogZ(p)
N(β,q,ε,XN,FN).(6.29)
Given σ(1),σ(2)∈ΣN, let
CN(σ(1),σ(2)):=
E
hXN(σ(1))XN(σ(2))i,e
CN(σ(1)):=CN(σ(1),σ(1)).
Define
VN:={CN(σ,σ):σ∈ΣN},V:=∞
[
N=1
VN.
The following result establishes a large deviations type relation between the partial free energy
and the full one.
Theorem 6.2.1. Assume XN={XN(σ)}σ∈ΣNis a centred Gaussian process and FN:ΣN→
R
are such that, for all N,M∈
N
, all σ(1),σ(2)∈ΣN, and all τ(1),τ(2)∈ΣM,
CN+M(σ(1)qτ(1),σ(2)qτ(2))≤N
N+MCN(σ(1),σ(2))+ M
N+MCM(τ(1),τ(2)),(6.30)
FN+M(σ(1)qτ(1))≤N
N+MFN(σ(1))+ M
N+MFM(τ(1)).(6.31)
Assume that CNand FNare bounded uniformly in N.
Then
6.2 Partial partition functions, external fields and overlaps 133
1. The following holds
pN(β,XN,FN):=1
Nlog ∑
σ∈ΣN
expβ√NXN(σ)+NFN(σ)
−−−→
N↑+∞p(β,X,F),almost surely and in L1.(6.32)
2. The limiting free energy p(β,X,F)is almost surely deterministic.
3. We have
lim
ε↓+0lim
N↑+∞p(p)
N(β,q,ε,XN,FN) =:p(p)(β,q,X,F)
=sup
v∈V
inf
λ∈
R
,γ∈
R
−λq−γv+p(β,X,λF+γe
C),
almost surely and in L1.(6.33)
4. Finally,
p(β,X,F) = sup
q∈Up(p)(β,q,X,F)+q.(6.34)
Remark 6.2.2.
1. If there exists {constN∈
R
+}∞
N=1such that, for all σ∈ΣN,
e
CN(σ) = constN,(6.35)
then (6.33) simplifies to
p(p)(β,q,X,F) = inf
λ∈
R
(−λq+p(β,X,λF)),almost surely and in L1.(6.36)
2. Inequality (6.31) can alternatively be substituted by the assumption (see (Guerra & Toninelli,
2003, Theorem 1)) that FN(σ) = f(SN(σ)), where f :
R
→
R
, f ∈C1(
R
), and SN:ΣN→
R
is the bounded function such that, for all σ∈ΣN,τ∈ΣM,
SN+M(σqτ) = N
N+MSN(σ)+ M
N+MSM(τ).
3. It is easy to check that the assumptions of Proposition 6.2.1 are fulfilled, e.g., for
XN:=c1GREMN+c2SK(p)
N,
and
FN(·):=f1(RN(·,σ(N)))+ f2(qL(·,σ(N))),
where σ(N)∈ΣN, c1,c2∈
R
, and f1,f2:
R
→
R
, such that f1∈C1(
R
), f2is convex. Note that
in this case, due to Proposition 6.2.1, the free energies (6.32) and (6.33) does not depend on
the choice of the sequence {σ(N)}∞
N=1⊂ΣN.
134 6 The GREM in the presence of uniform external field
Proof. Similarly to (Contucci et al., 2003, Theorem 1) and (Guerra & Toninelli, 2003, Theo-
rem 1) we obtain (6.32). Then (6.32) implies that
p(β,XN,λFN+γe
CN)−−−→
N↑+∞p(β,X,λF+γe
C),almost surely and in L1.
Hence, we can apply the quenched large deviation results (Theorems 3.3.1 and 3.3.2) which
readily yield (6.34) and (6.33) (or (6.36), in the case of (6.35)). ut
Remark 6.2.3. Derrida & Gardner (1986) sketched a calculation of the free energy defined in
(6.32) in the following case
FN=qLand XN=GREMN.(6.37)
This case is easier than the case (0.10) we are considering here, since both qLand GREMN
have lexicographic nature, cf. (0.5) and (0.2).
6.3 The REM with external field revisited
In this section, we recall some known results on the limiting free energy of the REM with
external field. However, we give some new proofs of these results which illustrate the approach
of Section 6.2. Moreover, we prove the weak limit theorem for the ground state and for the
partition function of the REM with external field.
Recall that the REM corresponds to the case n=1 in (1.30). This implies that the process X
is simply a family of 2Ni.i.d. standard Gaussian random variables. To emphasise this situation
we shall write REM(σ)instead of GREM(σ).
6.3.1 Free energy and ground state
Let us start by recalling the following well-known result on the REM.
Theorem 6.3.1 (Derrida (1980); Eisele (1983); Olivieri & Picco (1984)).Assume that n =1
and let p(β,h)be given by (6.1). The following assertions hold
1. We have
lim
N→∞pN(β,0) = (β2
2+log2,β≤√2log2
β√2log2,β≥√2log2 ,almost surely and in L1.(6.38)
2. For all β≥√2log2 and N ∈
N
, we have
0≤
E
[pN(β,0)] ≤βp2log2.(6.39)
See, e.g., (Bovier, 2006, Theorem 9.1.2) for a short proof.
Given k∈[0;N]∩
N
, define the set of configurations having a given magnetisation
ΣN,k:={σ∈ΣN:
N
∑
i=1
σi=N−2k}.(6.40)
6.3 The REM with external field revisited 135
Lemma 6.3.1. Set tk,N:=N−2k
N. Given any ε>0, uniformly in k ∈[0;N]∩
N
such that
tk,N∈[−1+ε;1 −ε],
we have the following asymptotics
N
k=
N↑+∞r2
π
2Ne−NI(tk,N)
qN(1−t2
k,N) 1+1
N 1
12 +1
3(1−t2
k,N)!+O1
N2!.(6.41)
Proof. A standard exercise on Stirling’s formula. ut
Theorem 6.3.2 (Dorlas & Wedagedera (2001)).Assume that n =1(the REM case) and let
p(β,h)be given by (6.1). We have
p(β,h):=lim
N→∞pN(β,h)
=(log2 +logchβh+β2
2,β≤p2(log2 −I(t∗)) =:β0
β(p2(log2 −I(t∗))+ht∗),β≥p2(log2 −I(t∗)) ,almost surely and in L1,
(6.42)
and t∗∈(−1;1)is a unique maximiser of the following concave function
(−1;1)3t7→ht +p2(log2 −I(t)).
Proof. For the sake of completeness, we give a short proof based on (the ideas of) Theo-
rem 6.2.1. Put
Mk,N:=(blog2N
kc,k∈[1;N−1]∩
N
1,k∈{0,N},
where bxcdenotes the largest integer smaller than x. Consider the free energy (cf. (6.29)) of the
REM of volume Mk,N
pk,N(β):=1
Mk,N
log ∑
σ∈Σk
N
expβM1/2
k,NREM(σ),
where REM :={REM(σ)}σ∈ΣNis the family of standard i.i.d. Gaussian random variables. Let
e
pk,N(β):=Mk,N
Npk,N N
Mk,N1
2β.(6.43)
Note that (6.43) is the restricted free energy (cf. (6.29)) of the REM, where the restriction is
imposed by the total magnetisation (6.3) given by tk,N.
We claim that the family of functions P:={
E
pN(·)}N∈
N
is uniformly Lipschitzian. In-
deed, uniformly in β≥0, we have
136 6 The GREM in the presence of uniform external field
∂β
E
pN(β)=N−1/2
E
GN(β,0)XN(σ)≤N−1/2
E
max
σ∈ΣN
X(σ)−−−→
N↑+∞p2log2.
Hence, the family Phas uniformly bounded first derivatives.
Given t∈(−1;1)and tkN,N∈UN(cf. (6.27)) such that limN↑+∞tkN,N=t, using (6.41), we
have
lim
N↑+∞
MkN,N
N=1−I(t)log2e.(6.44)
Using (6.44) and the uniform Lipschitzianity of the family P, we get
lim
N↑+∞e
pkN,N(β)=(1−I(t)log2e)p β
p1−I(t)log2e!.(6.45)
Combining (6.45) with (6.34) and (6.36), we get
p(β,h) = max
t∈[−1;1](tβh+(1−I(t)log2e)p β
p1−I(t)log2e!).(6.46)
To find the maximum in (6.46), we consider two cases.
1. If β≤p2(log2 −I(t∗)), then according to (6.38), we have
p β
p1−I(t)log2e!=log2 +β2
2(1−I(t)log2e).
Hence, (6.46) implies
p(β,h) = max
t∈[−1;1]tβh+β2
2+log2 −I(t)=log2 +logchβh+β2
2,(6.47)
where the last equality is due to the fact that the expression in the curly brackets is concave
and, hence, the maximum is attained at a stationary point. The stationarity condition reads
t=t0(β,h):=tanhβh.(6.48)
It is easy to check that the following identity holds
I(t) = ttanh−1t−logchtanh−1t.(6.49)
Combining (6.49) and (6.48), we get (6.47).
2. If β≥p2(log2 −I(t∗)), then again by (6.38), we have
p β
p1−I(t)log2e!=β√2log2
p1−I(t)log2e.
Hence, (6.46) transforms to
p(β,h) = max
t∈[−1;1]ntβh+βp2(log2 −I(t))o=βp2(log2 −I(t∗))+ ht∗,(6.50)
where the last equality is due to the concavity of the expression in the curly brackets.
6.3 The REM with external field revisited 137
Combining (6.47) and (6.50), we get (6.42). ut
Remark 6.3.1. We note that due to the continuity of the free energy as a function of β, we have
at the freezing temperature β0
t0(β0,h) = t∗(h).(6.51)
Theorem 6.3.2 suggests that the following holds.
Theorem 6.3.3. Under the assumptions of Theorem 6.3.2, we have
lim
N↑+∞
1
√Nmax
σ∈ΣN
XN(h,σ) = p2(log2 −I(t∗))+ht∗,almost surely and in L1.(6.52)
Proof. We have
1
βpN(β)≤1
NlogNβ√Nmax
σ∈ΣN
XN(h,σ)=logN
βN+1
√Nmax
σ∈ΣN
XN(h,σ).(6.53)
In view of (6.42), relation (6.53) readily implies that
p2(log2 −I(t∗))+ht∗≤lim
N↑+∞
N−1/2max
σ∈ΣN
XN(h,σ).(6.54)
We also have
1
βpN(β)≥1
√Nmax
σ∈ΣN
XN(h,σ)
which combined again with (6.42) implies that
p2(log2 −I(t∗))+ht∗≥lim
N↑+∞N−1/2max
σ∈ΣN
XN(h,σ).(6.55)
Due to the standard concentration of Gaussian measure (e.g., (Ledoux, 2001, (2.35))) and the
fact that the free energy (6.1) is Lipschitzian with the constant β√Nas a function of XN(h,·)
with respect to the Euclidean topology, the bounds (6.54) and (6.55) combined with the Borell-
Cantelli lemma give the convergence (6.52). ut
6.3.2 Fluctuations of the ground state
In this subsection, we shall study the limiting distribution of the point process generated by the
properly rescaled process of the energy levels, i.e. (6.15).
Proof of Theorem 6.1.1. Let us denote
EN(h):=∑
σ∈ΣN
δu−1
N,h(XN(h,σ)).(6.56)
We treat EN(h)as a random pure point measure on
R
. Given some test function ϕ∈C+
0(
R
)
(i.e., a non-negative function with compact support), consider the Laplace transform of (6.56)
corresponding to ϕ
138 6 The GREM in the presence of uniform external field
LEN(h)(ϕ):=
E
"exp(−∑
σ∈ΣN
ϕu−1
N,h(XN(h,σ)))#
=
N
∏
k=01
2πZ
R
exp−ϕu−1
N,h(x+h
√N(N−2k))−x2
2dx(N
k)
.(6.57)
Introduce the new integration variables y=u−1
N,h(x+h
√N(N−2k)). We have
(6.57) =
N
∏
k=0AN(h)
2πZ
R
exp−ϕ(y)−1
2uN,h(y)−h
√N(N−2k)2dy(N
k)
=expN
∑
k=0N
klog1−AN(h)
√2πZ
R
(1−e−ϕ(y))exp−1
2uN,h(y)−h
√N(N−2k)2.
(6.58)
Note that the integration in (6.58) is actually performed over y∈suppϕ, since the integrand is
zero on the complement of the support. It is easy to check that uniformly in y∈suppϕthe inte-
grand in (6.58) and, hence, the integral itself are exponentially small (as N↑+∞). Consequently,
we have
(6.58) =
N↑+∞exp−Zsuppϕ(1−e−ϕ(y))
N
∑
k=0N
kAN(h)
√2πexp−1
2uN,h(y)−h
√N(N−2k)2(1+o(1)).(6.59)
Denote tk,N:=N−2k
N. Using Lemma 6.3.1, we get
(6.59) =
N↑+∞exp−(1+o(1))Zsuppϕ(1−e−ϕ(y))
×
N
∑
k=0
AN(h)
πN(1−t2
k,N)1/2expNlog2 −I(tk,N)−1
2uN,h(y)−htk,N√N2.
(6.60)
Note that despite the fact that Lemma 6.3.1 is valid only for tk,N∈[−1+ε;1 −ε], we can still
write (6.60), since the both following sums are negligible:
0≤∑
k:tk,N∈([−1;−1+ε]∪[1−ε,1])
AN(h)
πN(1−t2
k,N)1/2exphNlog2 −I(tk,N)
−1
2uN,h(y)−htk,N√N2i≤KN exp(−LN),
and
0≤∑
k:tk,N∈([−1;−1+ε]∪[1−ε,1])N
kAN(h)
√2πexp−1
2uN,h(y)−htk,N√N2≤KN exp(−LN).
6.3 The REM with external field revisited 139
Consider the sum appearing in (6.60)
SN(h,y):=
N
∑
k=0
AN(h)
πN(1−t2
k,N)1/2expNlog2 −I(tk,N)−1
2uN,h(y)−htk,N√N2.
(6.61)
Introduce the functions fN,gN:[−1;1]→
R
as
fN(t):=I(t)+ 1
2uN,h(y)
√N−ht2−log2,
gN(t):=AN(h)
π(N(1−t2))1/2.
Note that definition (6.5) implies
I0(t∗) = hµ(t∗).(6.62)
A straightforward computation using (6.6), (6.7) and (6.62) gives
f00
N(t) = I00(t)+ h>0,(6.63)
f0
N(t∗) = −h
(2µ(t∗)N)2y+logI00(t∗)+h
4π(1−t2
∗)(log2 −I(t∗))N=OlogN
N,(6.64)
fN(t∗) = −1
Ny+1
2logI00(t∗) +h
4π(1−t2
∗)(log2 −I(t∗))N+o1
N.(6.65)
Hence, since (6.64) vanishes even after being multiplied by √N, (6.64) is negligible for the
purposes of the asymptotic Laplace principle. This readily implies that uniformly in y∈suppϕ
SN(h,y)∼
N↑+∞
NgN(t∗)
22πf00
N(t∗)
N1/2
expN fN(t∗).(6.66)
Using (6.63), (6.64) and (6.65) in the r.h.s. of (6.66), we obtain that uniformly in y∈suppϕ
SN(h,y)∼
N↑+∞exp(−y).(6.67)
Finally, combining (6.67) and (6.60), we obtain
lim
N↑+∞LEN(h)(ϕ) = exp−Z
R
1−e−ϕ(y)e−ydy.(6.68)
The r.h.s. of (6.68) is the Laplace transform of PPP(e−xdx,x∈
R
). Then a standard result implies
the claim (6.10). ut
140 6 The GREM in the presence of uniform external field
6.3.3 Fluctuations of the partition function
In this subsection, we compute the weak limiting distribution of the partition function under the
natural scaling induced by (6.8). Define
CN(β,h):=expβM(h)N+β
2µ(t∗)logI00(t∗)+h
4π(1−t2
∗)(log2 −I(t∗))N,(6.69)
DN(β,h):=ch−2/3(βh)expNlog2 +logchβh+β2
2,(6.70)
α(β,h):=β
µ(t∗).(6.71)
Theorem 6.3.4. If β>µ(t∗), then
ZN(β,h)
CN(β,h)
w
−−−→
N↑+∞Z
R
eα(β,h)xdP(1)(x).(6.72)
If β<µ(t∗), then
ZN(β,h)
DN(β,h)
w
−−−→
N↑+∞1.(6.73)
Proof. This is a specialisation of Theorem 6.1.3 which is proved in Section 6.4.2. ut
6.4 The GREM with external field
In this section, we obtain the main results of this chapter concerning the GREM with external
field. We prove the limit theorems for the distribution of the partition function and that of the
ground state.
6.4.1 Fluctuations of the ground state
As in the REM, we start from the ground state fluctuations (cf. Theorem 6.1.1). The following
is the main technical result of this section which shows exactly in which situations the GREM
with external field has the same scaling limit behaviour as the REM with external field.
Proposition 6.4.1. Either of the following two cases holds
1. If, for all l ∈[2;n]∩
N
,
log2 −I(t∗(θ−1/2
l,nh))
log2 −I(t∗(h)) <θl,n,(6.74)
then we have
∑
σ∈ΣN
δu−1
N,h(XN(h,σ))
w
−−−→
N↑+∞PPP(e−x,x∈
R
d).(6.75)
6.4 The GREM with external field 141
2. If, for all l ∈[2,...,n]∩
N
,
log2 −I(t∗(θ−1/2
l,nh))
log2 −I(t∗(h)) ≤θl,n,(6.76)
and there exists (at least one) l0∈[2;n]∩
N
log2 −I(t∗(θ−1/2
l0,nh))
log2 −I(t∗(h)) =θl0,n,(6.77)
then there exits the constant K =K(ρ,h)∈(0;1)such that
∑
σ∈ΣN
δu−1
N,h(XN(h,σ))
w
−−−→
N↑+∞PPP(Ke−x,x∈
R
).(6.78)
Remark 6.4.1. If condition (6.76) is violated, i.e., there exists l0∈[2;n]∩
N
such that
log2 −I(t∗(θ−1/2
l,nh))
log2 −I(t∗(h)) >θl,n,(6.79)
then the REM scaling (cf. (6.10),(6.8)) is too strong to reveal the structure of the ground state
fluctuations of the GREM. Theorem 6.1.2 shows how the scaling and the limiting object should
be modified to capture the fluctuations of the GREM in this regime.
Proof. 1. Denote Nl:=∆xlN, for l∈[1;n]. We fix arbitrary test function ϕ∈C+
K(
R
), i.e., a
nonnegative function with compact support. Consider the Laplace transform LEN(h)(ϕ)of
the random measure EN(h)evaluated on the test function ϕ.
LEN(h)(ϕ):=
E
"exp−∑
σ∈ΣN
(ϕ◦u−1
N,h)(XN(h,σ))#
=
E
"∏
σ∈ΣN
exp−(ϕ◦u−1
N,h)(XN(h,σ))#.(6.80)
Consider also the family of i.i.d standard Gaussian random variables
{X(σ(l),σ(2),...,σ(n))|l∈[1;n]∩
N
,σ(l)∈ΣNl,...,σ(n)∈ΣNn}.
Given l∈[1;n]∩
N
and y∈
R
, define
LN(l,v):=
E
h∏
σ(l)q...qσ(n)∈Σ(1−xl−1)N
exp−ϕ◦u−1
N,h(v+alX(σ(l))+
...+anX(σ(l),...,σ(n))+h(1−xl−1)√Nm(σ(l),...,σ(n)))i.(6.81)
We readily have
142 6 The GREM in the presence of uniform external field
LEN(h)(ϕ) = LN(1,0).(6.82)
Due to the tree-like structure of the GREM, for l∈[1;n−1]∩
N
, we have the following
recursion
LN(l,v) = ∏
σ(l)∈ΣNl
E
hLN(l+1,v+alX+h∆xl√Nm(σ(l)))i,(6.83)
where Xis a standard Gaussian random variable. Introduce the following quantities
YN(h,y,v,t,l):=uN,h(y)−h(1−xl−1)√Nt −v.
We claim that, for any l∈[1;n]∩
N
, uniformly in v∈
R
satisfying
v≤√NM(h)−δ−1−ql−1µ(h)−h(1−xl−1)t∗(θ−1/2
l,nh),
we have
logLN(l,v)∼
N↑+∞−AN(h)
q2π(1−ql−1)
(1−xl−1)N
∑
k=0(1−xl−1)N
k
×Z
R
1−e−ϕ(y)exph−1
2(1−ql−1)YN(h,y,v,tk,(1−xl−1)N,l)2idy.(6.84)
We shall prove (6.84) by a decreasing induction in lstarting from l=n.
2. The base of induction is a minor modification of the proof of Theorem 6.1.1. By the defini-
tion (6.81) and independence, we have
LN(n,v) =
Nn
∏
k=0
E
exph−(ϕ◦u−1
h,N)(anX+h∆xn√Ntk,Nn+v)i(Nn
k).(6.85)
For fixed k∈[0;Nn]∩
Z
,
E
hexp−(ϕ◦u−1
N,h)(anX+h∆xn√Ntk,N+v)i
= (2π)−1
2Z
R
dxexph−x2/2−(ϕ◦u−1
N,h)(anx+h∆xn√Ntk,N+v)i.(6.86)
We introduce in (6.86) the new integration variable
y:=u−1
N,hanx+h∆xn√Ntk,Nn+v.(6.87)
Using the change of variables (6.87), we get that the r.h.s. of (6.86) is equal to
AN(h)
√2πanZ
R
dyexp−1
2a2
n
YN(h,y,v,tk,Nn,n)2−ϕ(y).(6.88)
Combining (6.85) and (6.88), we get
6.4 The GREM with external field 143
LN(n,v) =
Nn
∏
k=0AN(h)
√2πanZ
R
dyexph−1
2a2
n
YN(h,y,v,tk,Nn,n)2−ϕ(y)i(Nn
k)
=
Nn
∏
k=01−AN(h)
√2πanZ
R
dy1−e−ϕ(y)exph−1
2a2
n
YN(h,y,v,tk,Nn,n)2i(Nn
k).
Define
VN(h,v,t,n):=AN(h)
√2πanZ
R
dy1−e−ϕ(y)exph−1
2a2
n
YN(h,y,v,t,n)2i.
Given any small enough δ>0, it straightforward to show that uniformly in v∈
R
such that
v≤√NM(h)−h∆xnt∗(hθ−1/2
n−1,n)−δ,
we have
LN(n,v) =
N↑+∞
Nn
∏
k=01−Nn
kVN(h,v,tk,Nn,n)1+O(e−CN).(6.89)
Indeed, we have
exph−1
2a2
n
YN(h,y,v,tk,Nn,n)2i≤exph−Nnlog2 −I(tn)i.
Next, using the fact that (1−e−ϕ(·))∈C+
K(
R
), we get for some C>0
Z
R
dy1−e−ϕ(y)exph−1
2a2
n
YN(h,y,v,tk,Nn,n)2i≤Cexph−Nnlog2 −I(tn)i.
Applying the elementary bounds
x−x2≤log(1+x)≤x,for |x|<1
2(6.90)
to
x:=−AN(h)
√2πanZ
R
dy1−e−ϕ(y)exph−1
2a2
n
YN(h,y,v,tk,Nn,n)2i,
and using the fact that, due to (6.41), there exists C>0 such that uniformly in k∈[1;Nn]∩
N
x2≤exph−2Nnlog2 −I(tn)iNn
k≤e−CN,
we get (6.89) and, consequently, (6.84) holds for l=n.
3. For simplicity of presentation, we prove only the induction step l=n l=n−1. Due to
(6.83), we have
LN(n−1,v) =
Nn−1
∏
kn−1=0
E
hLN(n,v+an−1X+h∆xn−1√Ntkn−1,Nn−1)i(Nn−1
kn−1).(6.91)
144 6 The GREM in the presence of uniform external field
Define
t(kn,kn−1):=1
1−xl−2∆xntkn,Nn+∆xn−1tkn−1,Nn−1.
Fix an arbitrary δ>0 and ε>0. Due to (6.84) with l=n, there exists some C>0, such
that uniformly for all kn,kn−1with
tkn,kn−1∈{t∈[−1;1]:|t∗(θ−1/2
n−1,nh)−tkn,kn−1|≤ε},
and uniformly for all v,x∈
R
satisfying
∆xn(log2 −I(tkn,Nn)) ≤1
2a2
nM(h)−δ−an−1x−N−1/2v
−h(∆xntkn,Nn+∆xn−1tkn−1,Nn−1)2,(6.92)
we obtain
logLN(n,v+an−1x+h∆xn−1√Ntk,Nn−1)≤CN exp(−N/C).(6.93)
Define
xN(v):=√N
an−1M(h)−δ−vN−1/2−an(2∆xn(log2 −I(tkn,Nn)))1/2
−h(∆xntkn,Nn+∆xn−1tkn−1,Nn−1).(6.94)
Using the elementary bounds
1+x≤ex≤1+x+x2,for |x|<1,(6.95)
and the bound (6.93), we obtain
E
h
1
{X≤xN(v)}LN(n,v+an−1X+h∆xn−1√Ntkn−1,Nn−1)i
=
N↑+∞
P
{X≤xN(v)}+
E
h
1
{X≤xN(v)}logLN(n,v+an−1X+h∆xn−1√Ntkn−1,Nn−1)i
+O(Nexp(−N/C)).(6.96)
Given kn−1∈[1;Nn−1]∩
N
, we have
E
1
{X≤xN(v)}exp−1
2a2
n
YN(h,y,v+an−1X+h∆xn−1√Ntkn−1,Nn−1,tkn,Nn,n)2
=1
√2πZxN(v)
−∞
dxexp−x2
2−1
2a2
nuN,h(y)−an−1x−h√N(∆xntkn,Nn+∆xn−1tkn−1,Nn−1)−v2
=exp−1
1−qn−2
YN(h,y,v,t(kn,kn−1),n−1)2
6.4 The GREM with external field 145
×1
√2πZxN(v)
−∞
exp
−a2
n+a2
n−1
2a2
n x−an−1
a2
n+a2
n−1
YN(h,y,v,t(kn,kn−1),n−1)!2
dx.
(6.97)
We claim that due to the strict inequalities (6.74), we have
1
√2πZxN(v)
−∞
exp
−a2
n+a2
n−1
2a2
n x−an−1
a2
n+a2
n−1
YN(h,y,v,t(kn,kn−1),n−1)!2
dx
−−−→
N↑+∞
an
a2
n+a2
n−11/2,(6.98)
uniformly in v∈
R
such that
v≤√NM(h)+δ0−h(∆xntkn,Nn+∆xn−1tkn−1,Nn−1)−(a2
n+a2
n−1)µ(h)=:vmax
N,(6.99)
where 0 <δ0exists due to strict inequality (6.74), for l=n. Indeed, due to the standard
bounds on Gaussian tails, to show (6.98) it is enough to check that
an−1
a2
n+a2
n−1
YN(h,y,v,t(kn,kn−1),n−1)+δ√N≤xN(v),(6.100)
for vsatisfying (6.99). Due to (6.74) with l=n, there exists δ3>0 such that we have
(2∆xn(log2 −I(tkn,Nn)))1/2≤µ(h)−δ3.(6.101)
Choosing a small enough δ0>0, we have
xN(v)−an−1
a2
n+a2
n−1
YN(h,y,v,t(kn,kn−1),n−1)+δ√N
=a2
n(M(h)−vN−1/2−h∆xntkn,Nn+∆xn−1tkn−1,Nn−1)
−(a2
n+a2
n−1)an(2∆xn(log2 −I(tkn,Nn)))1/2−δ
≥
(6.99)
a2
n(a2
n+a2
n−1)µ(h)−δ0−(a2
n+a2
n−1)an(2∆xn(log2 −I(tkn,Nn)))1/2−δ
≥
(6.101)
a2
n(a2
n+a2
n−1)µ(h)−δ0−(a2
n+a2
n−1)a2
nµ(h)−δ
= (a2
n+a2
n−1)(δ3a2
n+δ)−a2
nδ0>0
which proves (6.100).
We claim that there exists C>0 such that uniformly in kn−1∈[1;Nn−1]∩
N
and in v∈
R
satisfying (6.99) we have
Nn−1
kn−1
P
{X≥xN(v)}≤exp(−N/C).(6.102)
146 6 The GREM in the presence of uniform external field
Indeed, in view of (6.41) and due to the classical Gaussian tail asymptotics, to obtain (6.102)
it is enough to show that
Nn−1(log2 −I(tkn−1,Nn−1)) ≤1
2x2
N(vmax
N).(6.103)
Using (6.99) and (6.94), we obtain
xN(vmax
N) = N1/2
an−1(a2
n+a2
n−1)µ(h)−an(2∆xn(log2 −I(tkn,Nn)))1/2+δ0−δ.(6.104)
If n>2, then due to strict inequality (6.74), for l=n−2, there exists δ00 >0 such that we
have
(a2
n+a2
n−2)µ(h)−δ00 >(log2 −I(t∗(θ−1/2
l,nh)))(a2
n+a2
n−2)(∆xn+∆xn−1)1/2
≥(2a2
n−1∆xn−1(log2 −I(tkn−1,Nn−1)))1/2
+(2a2
n∆xn(log2 −I(tkn,Nn)))1/2,(6.105)
where the last inequality may be obtained as a consequence of Slepian’s lemma (Slepian,
1962). If n=2, then (6.105) follows directly from Slepian’s lemma. Combining (6.104) and
(6.105), we get (6.103). Note that (6.102), in particular, implies that
P
{X≥xN(v)}≤exp(−N/C).(6.106)
Given kn−1∈[1;Nn−1]∩
N
, denote
LN(n−1,v,kn−1):=
E
hLN(n,v+an−1X+h∆xn−1√Ntkn−1,Nn−1)i(Nn−1
kn−1)
Due to (6.106) and (6.96), we have
LN(n−1,v,kn−1) =
E
h(
1
{X≤xN(v)}+
1
{X>xN(v)})LN(n,v+an−1X+h∆xn−1√Ntkn−1,Nn−1)i(Nn−1
kn−1)
=1+
E
h
1
{X≤xN(v)}LN(n,v+an−1X+h∆xn−1√Ntkn−1,Nn−1)i
+O
P
{X≥xN(v)}+Nexp(−N/C)(Nn−1
kn−1).
Using (6.102) and the standard bounds (6.90) and (6.95), we get
LN(n−1,v,kn−1) = expNn−1
kn−1
E
h
1
{X≤xN(v)}logLN(n,v+an−1X+h∆xn−1√Ntkn−1,Nn−1)i
+O(Nexp(−N/C)).
Applying (6.98), (6.97), (6.84), for l=n, we obtain
logLN(n−1,v,kn−1) = −AN(h)
q2π(a2
n+a2
n−1)
Nn
∑
kn=0Nn
knNn−1
kn−1
6.4 The GREM with external field 147
×Z
R
1−e−ϕ(y)exph−1
2(a2
n+a2
n−1)YN(h,y,v,tkn,kn−1,n−1)2idy
+O(Nexp(−N/C)).
Finally, we arrive at
logLN(n−1,v) =
Nn−1
∑
kn−1=0
logLN(n−1,v,kn−1)
=−AN(h)
q2π(a2
n+a2
n−1)
Nn
∑
kn=0
Nn−1
∑
kn−1=0Nn
knNn−1
kn−1
×Z
R
1−e−ϕ(y)exph−1
2(a2
n+a2
n−1)YN(h,y,v,tkn,kn−1,n−1)2idy
+O(N2exp(−N/C))
=−AN(h)
q2π(a2
n+a2
n−1)
Nn+Nn−1
∑
k=0Nn+Nn−1
k
×Z
R
1−e−ϕ(y)exph−1
2(a2
n+a2
n−1)YN(h,y,v,tk,Nn+Nn−1,n−1)2idy
+O(N2exp(−N/C)).
4. Combining (6.82) and (6.84) for l=1, we obtain
LEN(h)(ϕ) = exp−Z
R
1−e−ϕ(y)SN(h,y)dy+o(1),(6.107)
where SN(h,y)is given by (6.61). Invoking the proof of Theorem 6.1.1, we get that
LEN(h)(ϕ)−−−→
N↑+∞exp−Z
R
1−e−ϕ(y)e−ydy
=LP(e−x)(ϕ).
This establishes (6.75).
5. The proof of (6.78) is very similar to the above proof of (6.75). The main difference is that
(6.98) does not hold. Instead, if (6.77) holds for l0=n, then we have
1
√2πZxN(v)
−∞
exp
−a2
n+a2
n−1
2a2
n x−an−1
a2
n+a2
n−1
YN(h,y,v,t(kn,kn−1),n−1)!2
dx
−−−→
N↑+∞
an
a2
n+a2
n−11/2
P
X<√N
an−1qa2
n+a2
n−1hM(h)−vN−1/2−(1−xn−2)ht∗(hθ−1/2
n,n)
−(a2
n+a2
n−1)µ(h)i,(6.108)
148 6 The GREM in the presence of uniform external field
uniformly in
v≤√NM(h)−(1−xn−2)ht∗(hθ−1/2
n,n)−(a2
n+a2
n−1)µ(h)−δ0.
The subsequent applications of the recursion (6.83) to (6.108) give rise to the constant
K(h,ρ)∈(0;1)in (6.78).
ut
Proof of Theorem 6.1.2. The existence of the r.h.s. of (6.18) follows from (Bovier & Kurkova,
2004a, Theorem 1.5 (ii)). It remains to show convergence (6.18) itself. We apply Proposi-
tion 6.4.1 to each coarse-grained block. Note that the assumption (6.74) of Proposition 6.4.1
is fulfilled, due to the construction of the blocks, cf. (6.11), (6.17). The result then follows from
(Bovier & Kurkova, 2004a, Theorem 1.2).
The representation of the limiting ground state (6.19) is proved exactly as in (Bovier &
Kurkova, 2004a, Theorem 1.5 (iii))). ut
6.4.2 Fluctuations of the partition function
In this subsection we compute the limiting distribution of the GREM partition function under the
scaling induced by (6.8). The analysis amounts to handling both the low and high temperature
regimes. The low temperature regime is completely described by the behaviour of the ground
states which is summarised in Theorem 6.1.2. The high temperature regime is considered in
Lemma 6.4.1 below.
Lemma 6.4.1. Assume l(β,h) = 0. Then
exp"−Nlog2 +logchβh+β2
2#ch2/3(βh)ZN(β,h)
w
−−−→
N↑+∞K(β,h),(6.109)
where K(β,h) = 1, if β¯
γ1(h)<1, and K(β,h)∈(0;1), if β¯
γ1(h) = 1.
Proof. We follow the strategy of (Bovier & Kurkova, 2004a, Lemma 3.1). By the very construc-
tion of the coarse graining algorithm (6.11), we have
e
θ1,k≤e
θ1,J1=¯
γ1(h)2,k∈[1;J1]∩
N
,
e
θ1,k<e
θ1,J1,k∈(J1;n]∩
N
.(6.110)
Assume β¯
γ1(h)<1. Hence, due to (6.110), we have
βe
θ1/2
1,k<1,k∈[1;n]∩
N
.(6.111)
Strict inequality (6.111) implies that there exists ε>0 such that, for all k∈[1;n]∩
N
,
β2−1
2(β−ε)2qk<xklog2 −I(t∗(h(xk/qk)1/2)).(6.112)
6.4 The GREM with external field 149
We have
E
ZN(β,h)=
N
∑
k=0N
kexpβhtk,NN+β2N
2=:SN(β,h).(6.113)
Note that due to (6.41)
SN(β,h)∼
N↑+∞
N
∑
k=0
gN(tk,N)expN f (tk,N),(6.114)
where
f(t):=log2 −I(t)+ βht +β2/2,
gN(t):=2
πN(1−t2)1/2
.
A straightforward computation gives
f0(t0) = βh−tanh−1(t0(β,h)) = 0
f00(t0) = −(1−t2
0)−1=−ch2(βh),
gN(t0) = 2
πN(1−t2)1/2
=2
πN1/2
ch(βh).
The asymptotic Laplace method then yields
SN(β,h)∼
N↑+∞ch−2/3(βh)expNlog2 +logchβh+β2
2.(6.115)
For p≤q, define
GREM(p,q)
N(σ(1),...,σ(q)):=
q
∑
k=p
akX(σ(1),...,σ(k)).
Consider the event
EN(σ):=nGREM(1,k)
N(σ(1),...,σ(k))<(β+ε)qk√N,
for all k∈[1;n]∩
N
o.
Define the truncated partition function as
Z(T)
N(β,h):=∑
σ∈ΣN
1
EN(σ)exphβ√NXN(h,σ)i.(6.116)
The truncation (6.116) is mild enough in the following sense
E
hZ(T)
N(β)i=SN(β,h)
P
nGREM(1,k)
N(σ(1),...,σ(k))<εqk√N,for all k∈[1;n]∩
N
o
150 6 The GREM in the presence of uniform external field
∼
N↑+∞
E
ZN(β,h).(6.117)
We write
ZN(β)
E
ZN(β)=Z(T)
N(β)
E
hZ(T)
N(β)i×
E
hZ(T)
N(β)i
E
ZN(β)+ZN(β)−Z(T)
N(β)
E
ZN(β)
=: (I) ×(II) +(III).
Due to (6.117), we get
(II) ∼
N↑+∞1,(III) L1
−−−→
N↑+∞0.
To estimate (I), we fix any δ>0, and use the Chebyshev inequality
P
{|(I) −1|>δ}≤δ
E
hZ(T)
N(β)i−2
VarhZ(T)
N(β)i.(6.118)
Expanding the squares, we have
VarhZ(T)
N(β)i=
E
hZ(T)
N(β)2i−
E
hZ(T)
N(β)i2
=
n
∑
p=1
∑
σ(1)q...qσ(k)∈ΣxkN
E
"expn2β√NGREM(1,p)
N(σ(1),...,σ(p))
+2βhxpmxpN(σ(1),...,σ(p))√No
×∑
σ(p+1)q...qσ(n),
τ(p+1)q...qτ(n)∈Σ(1−xp)N,
σ(p+1)6=τ(p+1)
expnβ√N
×GREM(p+1,n)
N(σ(1),...,σ(n))+GREM(p+1,n)
N(τ(1),...,τ(n))
+h(1−xp)√N(m(1−xp)N(σ(p+1)q...qσ(n))+m(1−xp)N(τ(p+1)q...qτ(n))o
×
1
EN(σ(1)q...qσ(n))
1
EN(τ(1)q...qτ(n))#.(6.119)
Hence, due to the independence, we arrive at
VarhZ(T)
N(β)i≤
n
∑
p=1
xkN
∑
k=0N
k
E
"expn2β√NGREM(1,p)
N(σ(1),...,σ(p))
+hxptk,N√No
1
nGREM(1,p)
N(σ(1),...,σ(p))<(β+ε)qp√No#
× (1−xp)N
∑
k=0(1−xp)N
k
E
"expβ√N(GREM(p+1,n)
N(σ(1),...,σ(n))
6.4 The GREM with external field 151
+h(1−xp)tk,(1−xp)N√N#!2
.(6.120)
Assume that Xis a standard Gaussian random variable. Using the standard Gaussian tail bounds,
we have
E
"exp2β√N(GREM(1,p)
N(σ(1),...,σ(p))+hxptk,N√N)
1
nGREM(1,p)
N(σ(1),...,σ(p))<(β+ε)qp√No#
=expnN2β2qp+βhtk,No
P
X≥(β−ε)pqpN
≤
N↑+∞
CexpN2β2qp+βhtk,N−1
2(β−ε)2qp.(6.121)
Similarly to (6.114), using (6.41) and (6.121), we have
xkN
∑
k=0N
k
E
"exp2β√N(GREM(1,p)
N(σ(1),...,σ(p))
+hxptk,N√N)
1
nGREM(1,p)
N(σ(1),...,σ(p))<(β+ε)qp√No#
≤
N↑+∞
C
xkN
∑
k=0
expNxp(log2 −I(tk,xpN))+ 2β2qp+2βhxptk,xpN−1
2(β−ε)2qp=:PN(p).
(6.122)
Using (6.41), we also obtain
(1−xp)N
∑
k=0(1−xp)N
k
E
"expβ√N(GREM(p+1,n)
N(σ(1),...,σ(n))
+h(1−xp)√Ntk,(1−xp)N#
≤
N↑+∞
C
(1−xp)N
∑
k=0
expnN(1−xp)(log2 −I(tk,(1−xp)N))+ 1
2(1−qp)β2
+βh(1−xp)tk,(1−xp)No=:e
PN(p).(6.123)
Combining (6.120), (6.122) and (6.123), we get
VarhZ(T)
N(β)i≤
N↑+∞
N
∑
p=1
PN(p)e
P2
N(p).(6.124)
For any p∈[1;n]∩
N
, we have the following factorisation
152 6 The GREM in the presence of uniform external field
E
hZ(T)
N(β)i=
xpN
∑
k=0xpN
k
E
"expβ√N(GREM(1,p)
N(σ(1),...,σ(p))+hxptk,xpN√N)
×
(1−xp)N
∑
k=0(1−xp)N
kexpβ√N(GREM(p+1,n)
N(σ(1),...,σ(n))
+h(1−xp)Ntk,(1−xp)N√N)
1
EN(σ(1)q...qσ(n))#.(6.125)
Hence, again similarly to (6.114), we obtain
E
hZ(T)
N(β)i∼
N↑+∞C
xpN
∑
k=0
expNxp(log2 −I(tk,xpN))+ 1
2qpβ2+βhxptk,xpN
×
(1−xp)N
∑
k=0
expnN(1−xp)(log2 −I(tk,(1−xp)N))+ 1
2(1−qp)β2
+βh(1−xp)tk,(1−xp)No=:QN(p)×e
PN(p).(6.126)
Denote
RN(p):=qpβ2+2xpmax
t∈[−1;1]{log2 −I(t)+βht}.
We observe that similarly to (6.115) we have
Q2
N(p)
exp(NR(p)) ∼
N↑+∞C.(6.127)
Combining (6.124), (6.126), (6.127) and (6.51), we get
(6.118) ≤
N↑+∞
C
n
∑
p=1
PN(p)
Q2
N(p)=C
n
∑
p=1
PN(p)/exp(NR(p))
Q2
N(p)/exp(NR(p)) ≤
N↑+∞
C
n
∑
p=1
PN(p)
exp(NR(p))
≤
N↑+∞
C
n
∑
p=1
expN(β2−1
2(β−ε)2)qp−(log2 −I(t0))xp
=C
n
∑
p=1
expN(β2−1
2(β−ε)2)qp−(log2 −I(t∗(h(xp/qp)1/2)))xp−−−→
N↑+∞0,
(6.128)
where the convergence to zero in the last line is assured by the choice of εin (6.112). Finally,
combining (6.118) and (6.128), we get
(I)
P
−−−→
N↑+∞1.
This finishes the proof of (6.109) in the case β¯
γ1(h)<1.
The case β¯
γ1(h) = 1 is a little bit more tedious and uses the information about the low
temperature regime obtained in Theorem 6.1.2 in the spirit of the proof of (Bovier & Kurkova,
2004a, Lemma 3.1). The lemma follows. ut
6.4 The GREM with external field 153
Proof of Theorem 6.1.3. The proof is verbatim the one of (Bovier & Kurkova, 2004a, The-
orem 1.7), where the analysis of the high temperature regime (Bovier & Kurkova, 2004a,
Lemma 3.1) is substituted by Lemma 6.4.1. The low temperature regime is governed by the
fluctuations of the ground state which are summarised in Theorem 6.1.2.
ut
6.4.3 Formula for the free energy of the GREM
Proof of Theorem 6.1.4. The L1convergence follows immediately from Theorem 6.1.3. Almost
sure convergence is a standard consequence of Gaussian measure concentration, e.g., (Ledoux,
2001, (2.35)), and the Borell-Cantelli lemma. ut
7
Some open problems and outlook
In this thesis, we studied two types of the large sums of strongly correlated exponentials. The
first type is a sum of hierarchically correlated random variables (the GREM with external field).
The second type is an infinitesimal sum of genuine non-hierarchically strongly correlated ran-
dom variables (the SK model with multidimensional spins). We were interested in the limiting
behaviour of such sums as the number of their summands and the effective dimension of the
correlation structure simultaneously tend to infinity.
For the GREM with external field, we provided an explicit expression for the free energy
(Theorem 6.1.4) and even obtained precise information about the fluctuations of the partition
function (Theorem 6.1.3). Our understanding of the SK model with multidimensional spins is
only at the level of free energy (cf. Question 0.0.1) and is substantially less elaborate. Below we
list some open questions concerning the latter model.
In view of the bounds on the free energy (of the SK model with multidimensional spins)
obtained using the generalised AS2scheme (Theorems 3.1.1 and 3.1.2, cf. also (3.22)), it is
natural to pose the following question.
Question 7.0.1 (Parisi-type formula, the AS2scheme version).Using the notations of Chapter 3
and, in particular, of Theorem 3.1.1, for which a priori distributions of spins do we have
p(β) = sup
U∈U
inf
(x,Q,Λ)f(x,Q,Λ,U),(7.1)
where the infimum runs over all x satisfying (3.7), all Λ∈Sym(d), all Qsatisfying both (3.6)
and “Hadamard squares” Assumption 3.1.2?
In view of the bounds on the free energy obtained using generalised Guerra’s scheme (see
Theorems 4.1.1 and 4.1.2), we pose the following version of Question 7.0.1.
Question 7.0.2 (Parisi-type formula, Guerra’s scheme version).Using the notation of Theo-
rem 4.1.1, for which a priori distributions of spins do we have (7.1), where the infimum in (7.1)
runs over all x satisfying (3.7), all Qsatisfying (3.6), and all Λ∈Sym(d)?
In view of the results on the Parisi-type formula in the case of multidimensional Gaussian a
priori distribution (Chapter 5), we pose the following question.
Question 7.0.3 (simultaneous diagonalisation scenario).Using the notations of Section 4.2.3,
for which a priori distributions of spins does (7.1) hold, where the infimum in (7.1) runs over all
x∈Q0
n(1,1), all Q∈Qdiag(U,O,d)∩Q0
n(U,d), all O ∈O(d), all n ∈
N
and all Λ∈Sym(d)?
156 7 Some open problems and outlook
In view of the conditional positive answer to Question 7.0.2 which we obtained by Tala-
grand’s methodology of a priori estimates (cf. Theorem 4.3.1), we pose the following question.
Question 7.0.4. For which a priori distributions of spins is Assumption 4.3.1 satisfied?
A related question (see Remark 4.3.8) seems to be the following one.
Question 7.0.5. For which a priori distributions of spins does the condition (4.75) hold?
We embedded an open problem of strict convexity of the Parisi functional posed by Panchenko
(2005a); Talagrand (2006c) into a more general setting in Section 4.2. In view of our partial re-
sult on strict convexity (see Theorem 4.2.4), this problem can be generalised and paraphrased
in the following way.
Question 7.0.6 (multidimensional Parisi functional).Given a piece-wise continuous x ∈Q(1,1)
and Q ∈Q(U,d), assume fx,Qis the solution of (4.39). For which terminal conditions g and
parameters Q is the mapping Q0(1,1)3x7→ fx,Q(0,0)∈
R
strictly convex?
Remark 7.0.2. The case of Question 7.0.6 with d =1essentially corresponds to the problem
posed by Panchenko (2005a); Talagrand (2006c).
Remark 7.0.3. It would, perhaps, also be useful to understand the behaviour of the mapping
Q(U,d)3Q7→ fx,Q(0,0)∈
R
.
A
Appendix
The general result of Guerra & Toninelli (2003) implies that the thermodynamic limit of the
local free energy (3.5) exists almost surely and in L1. The following existence of the limiting
average overlap is an immediate consequence of this.
Proposition A.0.1. We have
E
GN(β)⊗GN(β)VarHN(σ)−
E
HN(σ)HN(σ0)−−−→
N↑+∞C(β)≥0,
where C :
R
+→
R
+.
Proof. The free energy is a convex function of β(a consequence of the H¨
older inequality).
Hence, by a result of Griffiths (1964) the following holds
lim
N↑∞
d
dβ
E
pN(β)=d
dβ
E
[p(β)].
Proposition 3.2.4 implies
d
dβ
E
pN(β)=β
E
GN(β)⊗GN(β)VarHN(σ)−
E
HN(σ)HN(σ0).
ut
The following super-additivity result is an application of the Gaussian comparison inequali-
ties obtained in Subsection 3.2.3. Note that the result does not provide enough information for
the cavity-like argument of Aizenman et al. (2003).
Proposition A.0.2. For any V≡B(U,ε)⊂U, we have
N
E
pN(V)+M
E
[pM(V)] ≤(N+M)
E
pN+M(V)+(N+M)O(ε),
as ε↓+0.
Proof. Define the process YN,M:={Y(σ):σ=αqτ;α∈ΣN,τ∈ΣM}as follows
Y(αqτ):=N
N+M1/2
X(1)
N(α)+M
N+M1/2
X(2)
M(τ),
158 A Appendix
where X(1)and X(2)are two independent copies of the process X. Given some Gaussian process
{C(σ)}σ∈ΣN, let us introduce the functional ΦN(β)[C]as follows
ΦN,M(β)[C]:=
E
hlog µ⊗(N+M)h
1
ΣN(V)
1
ΣM(V)exp(β√N+MC)ii.
Now, set ϕ(t):=ΦN+M(β)h√tXN+M+√1−tYN,Mi. Applying Proposition 3.2.5, we get
d
dtϕ(t) = β2(N+M)
2
E
[G(t)⊗G(t)[
VarXN+M(σ(1))−VarYN,M(σ(1))
−CovhXN+M(σ(1)),XN+M(σ(2))i−CovhYN,M(σ(1)),YN,M(σ(2))iii.(A.1)
Note that we have
ϕ(0) = N
E
pN(V)+M
E
[pM(V)],
ϕ(1)≤(N+M)
E
pN+M(V),(A.2)
where the last inequality is due to the fact that, for all α∈ΣN(V)and all τ∈ΣN(V), we have
αqτ∈ΣN+M(V).
Moreover, for σ=αqτwith α∈ΣN(V)and σ∈ΣM(V)we have
VarXN+M(σ)−VarYN,M(σ) =
N
N+MRN(α,α)+ M
N+MRM(τ,τ)
2
2−N
N+MkRN(α,α)k2
2
−M
N+MkRM(τ,τ)k2
2=O(ε).
Also, due to convexity of the norm, we have
CovhXN+M(σ(1)),XN+M(σ(2))i−CovhYN,M(σ(1)),YN,M(σ(2))i
=
N
N+MRN(α(1),α(2))+ M
N+MRM(τ(1),τ(2))
2
2−N
N+MkRN(α(1),α(2))k2
2
−M
N+MkRM(τ(1),τ(2))k2
2≤0.
Applying R1
0dtto (A.1) and using the previous two formulae, we get the claim. ut
Notation Index
Sets, spaces
N
natural numbers, i.e., {1,2,...}
R
real axis, i.e., (−∞;+∞)
R
+nonnegative reals, i.e., [0;+∞)
R
+compactified nonnegative reals, i.e., [0;+∞]
R
[−∞;+∞)
S
d−1unit Euclidean sphere in
R
dwith centre at 0 ∈
R
d
B(x,r)Euclidean ball with centre at xand radius r∈
R
+
R
d×dreal d×dmatrices
Sym(d)symmetric d×dmatrices
Sym+(d)symmetric non-negative-definite d×dmatrices
O(d)orthogonal d×dmatrices
C(k)(D)k-times continuously differentiable functions f:D⊂
R
d→
R
C+
K(D)nonnegative functions f:D⊂
R
d→
R
+with compact support
Mf(D)finite measures on the metric space D
M1(D)probability measures on the metric space D
Λa finite index set
Λ0an infinite index set
Σa single spin configuration space
ΣΛ(or ΣN) whole system configuration space
ΣIsing Ising spins, ΣIsing :={−1;1}
(Ω,F,
P
)probability space of disorder
G0(β)set of all infinite-volume (DLR) Gibbs measures
Q(U,d)set of all c`
adl`
ag Sym+(d)-valued non-decreasing (in the sense of
quadratic forms) “paths” which start in 0 and end in matrix U, (4.17)
Q0(U,d)set of all piece-wise constant paths in Q(U,d)with finite (but arbitrary)
number of jumps
Q0
n(U,d)set of all piece-wise constant paths in Q0(U,d)with exactly n∈
N
jumps
E(I)set of all equivalence relations on the set I
Relations
ΛbΛ0set Λis the finite subset of the infinite set Λ0
160 Notation Index
xα
w
−−→
(α)y(xα)weakly converges to yalong the direction α
xα
P
−−→
(α)y(xα)converges to yin probability along the direction α
xα∼
(α)yα(xα)and (yα)are asymptotically equivalent along the direction α, i.e.,
limαyα
xα=1
xα=
(α)
O(yα)“the big-O notation”, rough asymptotic domination along the direction
α, i.e., limα|xα
yα|<+∞
A0 matrix A∈
R
d×dis nonnegative definite
AB B −A0
X∼Yrandom variables Xand Yare equidistributed
i∼
kjBolthausen-Sznitman equivalence relation, (2.27)
Π1Π2partition Π1is not finer than Π2
Operations
x:=y(y=:x) element xequals yby definition
xqyconcatenation of the vectors xand y
[x]kprojection of the vector x∈
R
non its first k(with k≤n) coordinates,
i.e., (x1,...,xk)∈
R
k
[σ]Aprojection of the vector σ∈ΣΛ0on ΣA, where A⊂Λ
bxclargest integer smaller than x
x∨ymaximum of xand y
x∧yminimum of xand y
N(·)normalisation operation, (2.18)
cardΛcardinality of the set Λ
diagxdiagonal matrix in
R
d×dinduced by the vector x∈
R
d
detAdeterminant of the matrix A∈
R
d×d
∇f(x)gradient of the function f:D⊂
R
d→
R
, i.e., (∂1f(x),...,∂df(x))
∇2f(x)Hessian of the function f:D⊂
R
d→
R
, i.e., (∂2
u,vf(x))d
u,v=1
∂x eF(x)directional (Gˆ
ateaux) derivative of F:X→
R
at the point xalong the
direction e
hx,yiEuclidean scalar product between the vectors x,y∈
R
d, i.e., ∑d
u=1xuyu
hA,Bitracial scalar product between the matrices A,B∈
R
d×d, i.e.,
∑d
u,v=1Au,vBu,v
µ⊗νproduct measure
µ⊗Λproduct measure, i.e., Ni∈Λµ
xacoordinate-wise raising of the vector x∈
R
dto the power a∈
R
Aaentry-wise raising of the matrix x∈
R
d×dto the power a∈
R
intDinterior of the set D
convDconvex hull of the set D
∂Dborder of the set D
Norms, metrics, valuations
1
D(x)indicator function of the set D
E
[X]expectation of the random variable X
Notation Index 161
µ[X]average of the function X with respect to the measure µ, i.e., RXdµ
Var[X]variance of the random variable X
Cov[X,Y]covariance between the random variables Xand Y
kxk2Euclidean norm of the vector x∈
R
d, i.e., qx2
1+···+x2
n
kAkFFrobenius (Hilbert-Schmidt) norm of the matrix A∈
R
d×d
kµkTV total variation norm of the finite (signed) measure µ, i.e., kµkTV =
supD|µ(D)|
dH(σ(1),σ(2))Hamming distance,
dL(α(1),α(2))lexicographic distance, (0.3)
R(σ(1),σ(2))overlap matrix, (3.3)
Q(α(1),α(2))ultrametric overlap matrix, (3.14)
q(i,j)limiting ultrametric overlap of i,j∈
N
, see (2.26)
S(ν|µ)relative entropy, (1.6)
I(·)Cram`
er Entropy, (6.4)
I∗(·)large deviations (good) rate function, (3.41)
pΛfree energy
P(·)Parisi functional, (5.10)
C S (·)Crisanti-Sommers functional, (5.14)
Processes, random variables, measures
µa priori measure on Σ
µΛ(or µN) a priori product measure on ΣΛ(or ΣN)
GΛ(or GN) finite volume Gibbs measure on ΣΛ(or ΣN), (1.1)
G0(β)infinite-volume (DLR) Gibbs measure
GΛ(β,σ(c))Gibbsian local specification on ΣΛ, given the external condition σ(c) ∈
Λ0\Λ, (1.12)
P
probability on the space of disorder
U(D)uniform distribution on the set D⊂
R
d
REM(·)Derrida’s REM process, (1.24)
GREM(·)Derrida’s GREM process, (0.5)
SK(·)Sherrington-Kirkpatrick process, (1.20)
PPP(D3x7→ f(x)∈
R
)Poisson point process with the density f:D⊂
R
d→
R
PD(x,a)Poisson-Dirichlet point process
ξRuelle’s probability cascade, (2.22)
ENpoint process of the GREM energy levels, (6.15)
HΛ(A,h;σ)spin glass Hamiltonian, (1.14)
Abbreviations
i.i.d. independent identically distributed
k-D k-dimensional
LLN law of large numbers
LDP large deviations principle
CLT central limit theorem
PDE partial differential equation
162 Notation Index
BSDE backward stochastic differential equation
HJB equation Hamilton-Jacobi-Bellman equation
c`
adl`
ag continue `
a droite, limite `
a gauche (right continuous with left limits)
RS replica symmetric
RSB replica symmetry breaking
FRSB full replica symmetry breaking
RPC Ruelle’s probability cascade
REM random energy model
GREM generalised random energy model
CREM continuous GREM
SK model Sherrington-Kirkpatrick model
EA model Edwards-Anderson model
RKKY model Ruderman-Kittel-Kasuya-Yoshida model
AS2scheme Aizenman-Sims-Starr scheme
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