Slepian models for Gaussian random landscapes

JHEP05(2020)142
Published o SISSA by Sp inge
Recei ed:Decembe 26, 2019
Re ised:Ap il 27, 2020
Accep ed:May 4, 2020
Published:May 27, 2020
Slepian models o Gaussian andom landscapes
Jose J. Blanco-Pillado,a,b Kepa Sousacand Mikel A. U kiolaa
aDepa men o Theo e ical Physics, Uni e si y o he Basque Coun y, UPV/EHU,
48080, Bilbao, Spain
bIKERBASQUE, Basque Founda ion o Science,
48011, Bilbao, Spain
cIns i u e o Theo e ical Physics, Facul y o Ma hema ics and Physics,
Cha les Uni e si y in P ague,
V Holesˇo iˇck´ach 2, P ague, Czech Republic
E-mail: [email p o ec ed],[email p o ec ed],
[email p o ec ed]
Abs ac : Phenomenologically in e es ing scala po en ials a e highly a ypical in gene ic
andom landscapes. We de elop he ma hema ical echniques o gene a e cons ained an-
dom po en ials, i.e. Slepian models, which can globally ep esen low-p obabili y ealiza-
ions o he landscape. We gi e analy ical as well as nume ical me hods o cons uc hese
Slepian models o cons ained ealiza ions o a ull Gaussian andom ield a ound c i ical
as well as in lec ion poin s. We use hese echniques o nume ically gene a e in an e icien
way a la ge numbe o minima a a bi a y heigh s o he po en ial and calcula e hei
non-pe u ba i e decay a e. Fu he mo e, we also illus a e how o use hese me hods by
ob aining s a is ical in o ma ion abou he dis ibu ion o obse ables in an in la iona y
in lec ion poin cons uc ed wi hin hese models.
Keywo ds: Cosmology o Theo ies beyond he SM, S ochas ic P ocesses, Supe s ing
Vacua
A Xi eP in : 1911.07618
Open Access,c
The Au ho s.
A icle unded by SCOAP3.h ps://doi.o g/10.1007/JHEP05(2020)142
JHEP05(2020)142
Con en s
1 In oduc ion 1
2 P elimina ies o Gaussian andom ields 3
3 Slepian models o cons ained Gaussian andom ields 4
3.1 Slepian models o c i ical poin s 5
3.2 Slepian models o in lec ion poin s 7
3.3 2D nume ical implemen a ion 9
4 Tunneling in a Gaussian andom landscape 9
4.1 S a is ics o he ins an on ac ion 13
4.1.1 Dependence wi h he heigh 13
4.2 App oxima ions o he calcula ion o he ac ion 14
4.2.1 Thin wall app oxima ion 14
4.2.2 S aigh -pa h app oxima ion 15
4.3 The lowes ac ion 16
4.3.1 Exi angle 16
4.3.2 Es ima ing he lowes ac ion 16
5 In la ion in a Slepian andom landscape 18
5.1 1D in lec ion poin in la ion 19
5.2 Nume ical in lec ion poin s in a 2D landscape 20
5.3 S a is ics o in la iona y pa ame e s 21
6 Summa y and conclusions 24
A Cons uc ion o Slepian models 27
A.1 In oduc o y ema ks and some p ope ies o Gaussian andom a iables 27
A.2 Condi ioned Gaussian andom ec o s 27
A.3 Gaussian andom ields 29
A.4 Use ul co ela ions 29
A.5 The Kac-Rice o mula and condi ioned Gaussian andom ields 31
A.6 Condi ioned Gaussian andom ield o a c i ical poin 31
A.6.1 Analysis o a condi ioned 2D Gaussian ield 35
A.6.2 Dis ibu ion o heigh s and eigen alues o he Hessian a a c i ical
poin 35
A.7 Condi ioned Gaussian andom ield o an in lec ion poin 36
A.7.1 P obabili y dis ibu ion o he in lec ion poin pa ame e s 38
B Nume ical implemen a ion and es s o he p obabili y dis ibu ions 38
B.1 Gene a ion o Gaussian andom ields: Ka hunen-Lo`e e expansion 38
B.2 Nume ical e alua ions o c i ical poin s 39
B.3 Nume ical e alua ions o in lec ion poin s 41
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1 In oduc ion
The low ene gy desc ip ion o many highe dimensional heo ies in ol e a la ge numbe
o ields (moduli ields) ha need o be s abilized. This is no mally achie ed by he
exis ence o a po en ial ha ixes he alues o hese ields o a local minimum o ha
po en ial unc ion. A ypical example o his p ocedu e can be ound in S ing Theo y
compac i ica ion scena ios. In pa icula , models o lux compac i ica ion ha e been shown
o lead o an eno mous se o possible 4dpo en ials ha can ha e many local minima.
The ypical numbe o moduli ields in hese cases is qui e la ge, eaching o en he o de
o a ew hund ed. This makes p ohibi i ely di icul o s udy hese po en ials in de ail
and one is o ced o look o simple models whe e he ield space has been unca ed o a
small subse o ields. Al e na i ely, one can y o s udy hese models by aking a mo e
s a is ical app oach, whe e he scala po en ial is ega ded as a andom ield whose sample
space is he se o 4dlow-ene gy e ec i e po en ials. These ideas ha e been pu sued in
ela ion o he s udy o he s abili y o c i ical poin s in hese po en ials in [1–3], as well
as he desc ip ion o cosmological models o he ea ly uni e se in [4–6].
In many o hese s udies one is in e es ed in pa icula poin s o he landscape such
as, o example, a minimum wi h some alue o i s cosmological cons an , o an in lec ion
poin wi h a pa icula se o condi ions in i s de i a i es necessa y o i o sus ain in la-
ion. Howe e , depending on he es ic ions imposed, i may be e y di icul o ob ain
an example o he po en ial wi h hese cha ac e is ics by p oducing andom ealiza ions
o he scala po en ial. Indeed, me as able de Si e acua and in la iona y poin s com-
pa ible wi h obse a ions a e e y a e in gene ic landscapes, wi h p obabili ies scaling as
P∼exp(−Np
), whe e N is he numbe o scala ields in he heo y, and p > 0 is a
numbe o o de one [7–12]. To ob ain ealiza ions wi h he desi ed p ope ies, one can
o cou se use a Taylo expansion a ound he poin in ques ion and ake in o accoun he
p obabili y dis ibu ion o i s coe icien s [13,14]. Howe e his becomes qui e compli-
ca ed as one inc eases he numbe o ields and he ield ange ha one is in e es ed in.1
Mo eo e , wi h his ype o p ocedu es i is no possible o cap u e co ec ly he global
p ope ies o he scala po en ial, which a e essen ial o s udy quan um decay p ocesses
in he landscape. He e we p esen a di e en s a egy o gene a e hese po en ials ha
locally will be cons ained o ha e a pa icula o m, bu ha globally will s ill ep esen
a ai h ul ealiza ion o he andom landscape, he so-called Slepian models [16].
Se e al di e en me hods ha e been sugges ed as a way o ep esen hese andom
po en ials in he landscape. In his pape we will concen a e on po en ials desc ibed by
Gaussian Random Fields (GRFs). This is based on he assump ion ha he 4dpo en ial
can be hough o as a sum o many di e en e ms, o classical and quan um o igin, coming
om he compac i ica ion mechanism ende ing he inal esul a Gaussian andom ield.
This ype o models ha e also been s udied in connec ion o he dis ibu ion o acua and
i s s abili y [9,17,18] as well as in la ion [13–15,19,20] in he landscape. As an illus a ion
o he ma hema ical echniques p esen ed he e o he cons uc ion o cons ained GRFs we
de elop Slepian models ha a e locally desc ibed by c i ical poin s (maxima, minima and
1Fo ano he me hod o gene a ing a speci ic class o cons ained Gaussian andom ields, see [15].
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JHEP05(2020)142
saddle poin s) as well as in lec ion poin s and use hese ealiza ions o ex ac impo an
s a is ical in o ma ion abou hem.
In pa icula , we will i s s udy he quan um mechanical s abili y o local minima
in hese landscapes. In o de o do so, we will compu e nume ically he decay a e o
hese minima using he quan um unneling echniques i s desc ibed in a se ies o pape s
in [21,22]. The esul o his quan um ins abili y is he c ea ion o a bubble ins an on ha
in e pola es be ween he alse acuum and he ue acuum s a es. Using hese Euclidean
me hods one can e alua e he p obabili y o his decay channel and he e o e es ima e he
li e ime o any speci ic acuum. The calcula ion o hese unneling e en s in a mul idimen-
sional po en ial is howe e no o iously di icul . Recen ly some wo k on his di ec ion has
been done in ela ion o he s abili y o acua in models wi h la ge numbe o dimensions
in ield space. I has been a gued ha he p obabili y o he decay depends exponen ially
on he numbe o ields al hough he pa icula scaling is s ill unce ain [23–26].
In his pape we will s udy hese unneling e en s in models o Gaussian andom
po en ials. In pa icula we a e in e es ed in s udying he dependence o he unnelling a e
wi h he heigh o he po en ial a he alse acuum. Fo la ge alues o he cosmological
cons an his calcula ion would be impossible wi hou cons aining me hods, since he
numbe o hese minima is negligible compa ed o he minima a lowe alues o he ield.
Ou echniques allowed us o e icien ly gene a e he same numbe o minima o di e en
heigh s and ha e a good sample o cases om whe e we can ex ac s a is ical in o ma ion.
The ob ained dis ibu ion o he ins an on ac ions SE(which de e mines he decay a e
Γ∼e−SE) is displayed in igu es 5and 6, whe e we ound ha he a e age dependence o
he decay a e on he alse acuum heigh V is gi en by
log10 SE
U−1
0Λ4≈3.29 exp −0.18V
U0,
whe e U0and Λ a e he cha ac e is ic ene gy and leng h scale (in ield space) o ou po en ial
espec i ely. The dis ibu ion o he Euclidean ac ion becomes inc easingly peaked a ound
i s mean, and hus mo e p edic i e, o la ge alues o V . As we show in he main ex ,
his enhancemen o he p edic abili y can be explained using Slepian models o e y
a ypical ex ema o he po en ial, such as high minima.
Ou second applica ion in ol es he gene a ion o in lec ion poin s. These a e some o
he mos likely poin s in he landscape whe e cosmological in la ion can happen. Howe e
his does no mean ha an a bi a y in lec ion poin would lead o in la ion. Ob aining
a success ul in la iona y pe iod consis en wi h he cu en cosmological obse a ions s ill
equi es some amoun o ine uning o he po en ial a ound he in lec ion poin . The e o e,
o cha ac e ise he dis ibu ion o obse ables o hese in la iona y models in he landscape
one should again use some so o cons aining me hod, and look a a pa icula se o
non-gene ic in lec ion poin s. In he p esen pape we will explo e he dependence o he
obse able pa ame e s o in la ion o i s ini ial condi ions in he landscape. In pa icula we
will ake he ini ial condi ions o he ields o be he ones de e mined by he exi poin o an
ins an on desc ibing he ansi ion om a nea by pa en alse acuum. No e ha in o de
o pe o m his analysis, one equi es no only he knowledge o he po en ial a ound he
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JHEP05(2020)142
in lec ion poin bu also i s ela ion o nea by minima. Hence ou me hod, which accu a ely
cap u es he global s a is ical p ope ies o he po en ial, is pa icula ly sui able o ca y
ou his in es iga ion. I is wo h no ing ha , o he bes o ou knowledge, his is he i s
ime ha an Slepian model o in lec ion poin s is p esen ed in he li e a u e. The e ec
o he unneling in he ini ial s ages o in la ion has also been discussed in [14,27–29].
The emaining o he pape is o ganized as ollows. In sec ion 2we in oduce he
no a ion ha we will be using o desc ibing ou andom po en ial unc ion as a GRF.
In sec ion 3we will ou line he me hod o gene a ing cons ained andom po en ials as
Slepian models. In sec ion 4, we implemen hese ideas o a 2d ield space landscape
and gene a e a la ge se o andom po en ials wi h a minimum a a speci ic poin in ield
space. This allows us o compu e he unneling pa hs om hese minima and de e mine
he s a is ics o he decay a e. In sec ion 5, we condi ion he andom po en ial o ha e
an in lec ion poin sui able o in la ion, and s udy he e ec o he ini ial condi ions
se by he unneling p ocess om a nea by minimum. We conclude in sec ion 6wi h
some commen s on he esul s and some u he ideas ha can be implemen ed wi h hese
nume ical echniques. Some o he ma hema ical de ails and nume ical p oo s ha e been
le o he appendices. In he p esen wo k, unless o he wise s a ed, we will use educed
Planck uni s M−2
pl = 8πG/(~c) = 1.
2 P elimina ies o Gaussian andom ields
In his pape we will ake ou andom po en ial, V(φ), o be a Gaussian andom ield
de ined o e a N-dimensional ield space, which we will pa ame ize wi h he ec o φ=
{φi}, wi h i= 1, . . . , N. Fu he mo e, we will conside he p obabili y dis ibu ion o
he andom po en ial o be homogeneous and iso opic, so i s co a iance unc ion will only
depend on he dis ance be ween he poin s a which i is e alua ed, in o he wo ds i is o
he o m
hV(φ1)V(φ2)i=C(|φ1−φ2|).(2.1)
We will addi ionally equi e he po en ial o ha e a null mean:
hV(φ)i= 0 .(2.2)
In he es o he pape we will e alua e ou exp essions using he ollowing simple
co a iance unc ion:
C(φ) = U2
0exp −φ2
2Λ2,(2.3)
o he case o N= 2 ield space dimensions. The pa ame e U0se s he ene gy scale
o he po en ial while Λ ep esen s he co ela ion leng h in ield space. I is impo an
o ealize ha he echniques used in his pape a e gene ic and can be applied o o he
in e es ing si ua ions like, o example, non-Gaussian co a iance unc ions so in his sense
hese cons uc ions a e qui e mo e gene ic han he ones p esen ed in [15]. We ha e decided
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JHEP05(2020)142
o use he simple Gaussian co a iance unc ion since i conside ably simpli ies some o he
exp essions in his pape .2
In he ollowing we will be in e es ed in he alue o he ield and i s de i a i es a a
pa icula poin in ield space, which we can ake o be φ= 0 wi hou loss o gene ali y,
and we will e e o i as he cen e o ield space. In o de o simpli y he no a ion we
in oduce he ollowing de ini ions o he alue o he po en ial and i s de i a i es:
u=V(φ)|φ=0, ηi=∂V (φ)
∂φiφ=0
, ζij =∂2V(φ)
∂φi∂φjφ=0
, ρijk =∂3V(φ)
∂φi∂φj∂φkφ=0
.
Fu he mo e, we will deno e he eigen alues o he Hessian ma ix by λiwi h i= 1,2
which will single ou he di ec ions 1,2 in ou ield space. No e ha he de i a i es o
he scala po en ial a e also Gaussian andom a iables, and he e o e any collec ion o
he p e ious quan i ies o ms a Gaussian andom ec o . In appendix A.4 we will gi e
he exp essions o he co ela o s be ween hese di e en de i a i es o he po en ial as a
unc ion o he de i a i es o he co a iance unc ion C(φ). These co ela ions will play an
impo an ole in some pa s o ou discussions.
3 Slepian models o cons ained Gaussian andom ields
A key poin in ou cons uc ion o he GRF es s on he ac ha a condi ioned GRF
main ains i s Gaussian na u e. Mo e speci ically, homogeneous and iso opic p ocesses
(such as he GRFs we a e dealing wi h) can be condi ioned using he Kac-Rice o mula [30]
in o de o ob ain new mean and co a iance unc ions which gene a e GRFs wi h he
equi ed cons ain s.3The models o s ochas ic p ocesses dealing wi h condi ional e en s
and c ossings whe e pionee ed by Da id Slepian [16], and ha e hus been coined in he
ma hema ical li e a u e as Slepian models.
We can desc ibe hese cons ained p ocesses in a gene ic o m in he ollowing way.
Fo simplici y, le us conside i s a Gaussian andom p-dimensional ec o , composed
o join ly Gaussian a iables, xT= (x1, . . . , xp), whose p obabili y dis ibu ion unc ion
(PDF) is gi en by,
(x) = 1
(2π)p/2√de Σ exp −1
2(x−µ)TΣ−1(x−µ)(3.1)
whe e µ=hxiis he mean ec o and Σ is he co a iance ma ix, whose elemen s a e gi en by
Σab =h(xa−µa)(xb−µb)i,(3.2)
wi h a, b = 1, . . . , p.
Le us now conside he ollowing decomposi ion o he andom ec o x= (x1,x2),
whe e x2a e pccomponen s o he ec o x ha will be cons ained by a condi ion x2=˜x,
2No e, howe e , ha his co a iance unc ion leads o a somewha special o m o he Hessian ma ix
o he minima in his GRF (See o example he discussion o his poin in [18].) I would be in e es ing
o check whe he his could ha e any quan i a i e e ec on he conclusions o ou pape .
3See a b ie desc ip ion o he Kac-Rice o mula in he cu en con ex in appendix A.5.
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JHEP05(2020)142
and x1a e he emaining p−pcuncons ained elemen s. Then one can show [30,31] ha
he dis ibu ion p obabili y o x1holding x2 ixed o he desi ed alues is gi en by,
˜
(x1|x2=˜
x) = 1
(2π)p−pc
2pde ˜
Σ
exp −1
2(x1−˜
µ)T˜
Σ−1(x1−˜
µ),(3.3)
which shows ha he dis ibu ion o he a iables x1is indeed a Gaussian dis ibu ion
bu now wi h a mean and co a iance unc ions gi en in e ms o he o iginal ones as
˜µ=µ1+ Σ12Σ−1
22 (˜x−µ2),˜
Σ=Σ11 −Σ12Σ−1
22 Σ21 ,(3.4)
whe e µ1and µ2a e he means o he ec o s x1and x2 espec i ely, and
Σ11 =h(x1−µ1)(x1−µ1)i,
Σ12 = Σ21 =h(x1−µ1)(x2−µ2)i,
Σ22 =h(x2−µ2)(x2−µ2)i.(3.5)
This is possible because one can always ind a new Gaussian andom ec o
x0= (x0
1,x0
2), connec ed o he o iginal one wi h a non-singula linea ans o ma ion
x0=A·x, such ha x0
2=x2is unco ela ed o x0
1. We show in appendix A.2 a p oo o
his s a emen . In he es o he pape we will use his ac in se e al di e en ways, apply-
ing his echnique o Gaussian andom ec o s made o di e en quan i ies o ou po en ial.
3.1 Slepian models o c i ical poin s
In his sec ion we will use he me hods desc ibed ea lie o gene a e a Gaussian andom
ield wi h a c i ical poin wi h a speci ic heigh a he cen e , φ=0. In o he wo ds, we will
ind a desc ip ion o he new GRF condi ioned so ha he poin a i s cen e sa is ies he
ollowing p ope ies: V(0) = uand V0
i(0) = ηi= 0 o i= 1,2. In o de o do his we will
ollow he p esc ip ion used in he ma hema ical li e a u e o maxima in GRF [32] and
adap i o ou case. Le us s a by in oducing he ollowing Gaussian andom ec o :
x={V(φ1), . . . , V (φq), V (0), η1, η2, ζ11, ζ22, ζ12}(3.6)
whe e we deno e by φa, wi h a= 1, . . . , q, he posi ion in ield space o a disc e e se
o qpoin s. One can show ha he Gaussian andom ec o xhas ze o mean, and a
p obabili y dis ibu ion ha can be eadily compu ed using he o m o he co a iance
unc ion and i s de i a i es. This is a somewha leng hy calcula ion and we ha e gi en he
gene al exp ession in appendix A.6. Acco ding o he desc ip ion o cons ained Gaussian
andom ec o s gi en abo e his is all we need o ob ain he new mean and co a iance
unc ion o he new condi ioned ec o (and hus, also o he cons ained GRF).
Using he esul s in appendix A.6, one can show ha he new mean unc ion o he
GRF wi h he cons ained condi ions is gi en by,
˜µ(φ) = e−φ2
2Λ2"u1 + φ2
2Λ2+1
2
2
X
i=1
φ2
iλi#.(3.7)
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JHEP05(2020)142
This esul co esponds o he pa icula choice o co a iance unc ion in eq. (2.3), and
is w i en in e ms o he alue o he ield V(0) = uand he eigen alues o he Hessian
ma ix a he cen e , λi, which a e o be d awn om he dis ibu ion in eq. (3.9) below.
The new co a iance unc ion is
˜
C(φ1,φ2) = U2
0exp −|φ1|2+|φ2|2
2Λ2exp φ1·φ2
Λ2−1−φ1·φ2
Λ2−(φ1·φ2)2
2Λ4,
(3.8)
which is no longe homogeneous, bu i is s ill iso opic.
I is impo an o no e ha he eigen alues o he Hessian a e no s a is ically indepen-
den o he heigh o he po en ial. This is in ui i ely clea since, o example, one would
expec he ypical minimum a a la ge heigh o be qui e shallow compa ed o he minima
si ua ed well bellow he mean alue o he po en ial. This expec a ion can be ansla ed
o he exis ence o impo an co ela ions be ween he ield and i s second de i a i es a a
poin , and in pa icula a c i ical poin s. In o de o ake his e ec in o accoun one can
calcula e he join p obabili y dis ibu ions o he Hessian eigen alues (λi) and heigh s (u)
a c i ical poin s o ob ain4
Pu,λ du Y
i
dλi=Nexp −u2
2U2
0|λ1−λ2|
2
Y
i=1 |λi|exp "−Λ2λi+u
2U02#dλidu , (3.9)
whe e Nis a no malizing cons an . This dis ibu ion includes all ypes o c i ical poin s,
namely maxima, minima and saddle poin s. Depending on he kind we a e in e es ed in,
we simply need o impose posi i i y o nega i i y condi ions on he alues o each λi.
Using hese esul s we can gene a e a Gaussian andom ield wi h a c i ical poin wi h
he desi ed p ope ies by he ollowing p ocedu e. Le us conside o example a minimum
wi h ixed heigh u. Ou i s s ep will be o gene a e a se o eigen alues d awn om he
dis ibu ion (3.9) aking in o accoun he alue o u, imposing he non-nega i i y condi ion
λi≥0, and ixing he no maliza ion ac o acco dingly.
Using hese alues o λiwe can hen gene a e ealiza ions o he po en ial using he
exp ession
V(φ) = e−φ2
2Λ2"u1 + φ2
2Λ2+1
2
2
X
i=1
φ2
iλi#+ ∆(φ) (3.10)
whe e we ha e deno ed by ∆(φ) an inhomogeneous, ze o-mean Gaussian andom ield
whose co a iance unc ion is gi en by ˜
C(φ1,φ2) in eq. (3.8). We show in igu e 1an
example o he di e en ing edien s ha make up a Slepian model o a local minimum
in a 1dGRF. We can use a simila p ocedu e o gene a e o he c i ical poin s, such as
saddle poin s wi h di e en numbe o nega i e eigen alues, by gene a ing he app op ia e
samples o λi’s.
An impo an conclusion ha can be de i ed om he Slepian model (3.10), i s
no iced in [32], is ha o highly non-gene ic ex ema |u|  U0(such as e y low maxima
o high minima), he shape o his GRF becomes e y de e minis ic a ound he c i ical
4See he calcula ion in appendix A.6.
– 6 –
JHEP05(2020)142
μ(ϕ)
Δ(ϕ)
-4-2 2 4 ϕ
-2.0
-1.5
-1.0
-0.5
0.5
-4-2 2 4
ϕ
-2.0
-1.5
-1.0
-0.5
0.5
V(ϕ)
Figu e 1. A 1d example o a Slepian model o a cons ained minimum in a GRF. We show, o
a pa icula ealiza ion, he wo sepa a e componen s o he cons uc ion on he le , namely, he
cons ained mean ield o m µ(φ) in eq. (3.7) and he inhomogenous new GRF ∆(φ) wi h co a iance
unc ion gi en by eq. (3.8). The o al GRF is shown on he igh .
poin , and i is desc ibed e y accu a ely by he i s wo e ms in eq. (3.10). Indeed, one
can see om eq. (3.8) ha he s anda d de ia ion o he andom componen ∆(φ) is always
smalle han U0, and ha i app oaches ze o nea he ex emum loca ed a φ=0(see also
igu e 1). The e o e he las con ibu ion in (3.10) can be neglec ed in a neighbou hood
o he ex emum whe e |∆(φ)|.U0 |V(0)|holds. On he o he hand, in he limi
|u|  U0 he eigen alue dis ibu ion o he Hessian (3.9) is app oxima ely gi en by5
Pλdλ1dλ2∼ |λ1−λ2||λ1||λ2|exp −Λ2|(λ1+λ2)u|
2U2
0dλ1dλ2,(3.11)
which indica es ha in his limi he magni ude o he eigen alues is e y supp essed
|λi|  U0/Λ2. Then, as we men ioned abo e, o highly non-gene ic ex ema he decom-
posi ion (3.10) is domina ed by i s de e minis ic pa ( he i s e m), wha makes hese
Slepian models e y p edic i e in hose si ua ions. As we shall see bellow, his esul is
pa icula ly impo an when we conside he dis ibu ion o non-pe u ba i e decay a es
om minima wi h a la ge acuum ene gy. Fo an example o a ealiza ion wi h a high
minimum see igu e 2(a).
This de e minis ic cha ac e o la ge luc ua ions o Gaussian Random Fields plays an
impo an ole in a ious a eas o Cosmology, such as he analysis o he CMB da a, and
he s udy o La ge Scale S uc u e o ma ion in he uni e se (see e.g. [33–37]).
3.2 Slepian models o in lec ion poin s
As we discussed in he In oduc ion, we a e also in e es ed in in lec ion poin s in he
landscape. The eason is ha in a cosmological con ex hese poin s could be one o he
egions o he po en ial ha gi e ise o a cosmological in la iona y pe iod. Howe e , in
o de o be compa ible wi h he la es cosmological obse a ions, one needs o es ic he
o m o hese in lec ion poin s. This leads us o conside an in lec ion poin a φ= 0 as a
ealiza ion o he GRF wi h a small g adien o he po en ial in he φ1di ec ion, deno ed
by η1, and he es o he coe icien s o he Taylo expansion o he ield a ound ha poin
5No e ha o e y high minima u > 0 and λi>0, while o e y low maxima all signs a e e e sed.
– 7 –
JHEP05(2020)142
▲
▲
▲
▲
▲▲▲▲
■
■
■
■■■■■
Op imal pa h
▲
Linea app oxima ion
■Thin-wall app oxima ion
-2-1 0 1 2 3 4 5 V(ϕFV)/U0
100
103
104
105
S
Figu e 6. E olu ion o he median o he ac ion, wi h e o ba s ep esen ing da a be ween he
i s and hi d qua iles o each dis ibu ion, o he op imal pa h, he linea (s aigh -pa h) and
he hin wall app oxima ions, along wi h a i ing cu e (see (4.6)).
o (4.6) o he expec ed alue o he ac ion. This enhancemen o he p edic abili y o he
Slepian model o la ge alues o V co esponds p ecisely o wha we an icipa ed in he
p e ious sec ion. Indeed, he e we showed ha nea high minima he andom po en ial
becomes domina ed by he i s e m in he decomposi ion (3.10), and he e o e he land-
scape is e y de e minis ic in a neighbou hood o alse acua wi h la ge V . Consis en
wi h his esul , when s udying he non-pe u ba i e s abili y om hese acua we obse e
a educ ion o he a iance o unneling ac ions o la ge heigh s o he alse acuum. This
ag eemen also sugges s ha in he case o minima wi h a la ge V he alue o he ins an-
on ac ion is domina ed by he local s uc u e o he minimum. We will p o ide u he
e idence o his claim below.
4.2 App oxima ions o he calcula ion o he ac ion
Due o he inhe en ins abili y o he equa ions o be sol ed o compu e unneling p o iles,
i is clea ha as we inc ease he domain and dimensionali y o he po en ial unde s udy,
he equi ed compu a ional ime o sol e he sys em will g ow acco dingly. E iden ly,
his makes he s udy o highe -dimensional GRFs and hei unneling p ope ies almos
p ohibi i e in his sense. Mo i a ed by hese limi a ions, we u n o compu ing se e al
di e en app oxima ions o unneling ac ions sugges ed in he li e a u e, and compa e
hem wi h ou exac esul s.
4.2.1 Thin wall app oxima ion
The hin-wall p esc ip ion was al eady discussed in he o iginal pape s by Coleman in [21].
In his app oxima ion he ins an on ac ion is gi en in e ms o he di e ence be ween
po en ial a he alse acuum (V ) and ue acuum (V ) and σ, he ension o he wall
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JHEP05(2020)142
in e pola ing be ween hem, namely,
¯
S w =27π2σ4
2(V −V )3, σ =ZφF V
φT V
dφ p2(V(φ)−V(φTV )) .(4.7)
This app oxima ion is accu a e as long as he di e ence be ween V and V is small.
We e alua ed (4.7) o each bounce we p e iously ound wi h AnyBubble in o de o
check his exp ession and i s p edic i e powe o GRFs. In he compu a ion we es ic ed
he ield o a s aigh line in ield space connec ing he ue and alse acua. Figu e 6shows
he e olu ion o he median o ¯
S w as a unc ion o he alse- acuum heigh . While he wid h
and median o he dis ibu ion in his case ollow he same pa e n as he op imal ac ion,
he alues di e ge apidly om he op imal ones as he alse acuum heigh inc eases. This
is no oo su p ising since, as one inc eases he heigh o he alse acuum minimum, he
ield can unnel o a minimum wi h qui e di e en alues o he po en ial, wha iola es
one o he p emises o he hin wall app oxima ion.
4.2.2 S aigh -pa h app oxima ion
While he hin-wall p esc ip ion p o ides a solid uppe bound on he bounce ac ion [43],
i does no p o ide any use ul es ima ion on he ac ual alue on he bounce in ou case.
This ac calls o an al e na i e way o es ima e he ac ion, mos ly o highe -dimensional
landscapes.
A s aigh o wa d simpli ica ion o his p oblem was in oduced in [44], which we
will deno e by s aigh -pa h app oxima ion. This p esc ip ion is based on educing he ield
space o a single s aigh line connec ing he alse and ue acua, hus making he p oblem
o unneling e ec i ely one-dimensional. As can be seen om igu e 3, his app oxima ion
may no be oo un easonable. E en hough he e a e some pa hs which do cu e o e he
ield space, many (i no mos ) o hem ollow a s aigh ajec o y in ield space. No e,
howe e , ha his es ic ion in ield space may yield e ec i e po en ials whe e he bounce
does no exis o migh e en co espond o a di e en bounce in he ull heo y. Fo mo e
de ails on he p ope ies o his app oxima ion, see [45].
Fo each op imal pa h, we conside ed a s aigh line in he wo-dimensional GRF
connec ing he ue and alse acua, and compu ed he co esponding es ima e o he ac ion,
¯
Ssp, in each case. In p inciple, ¯
Ssp ep esen s an uppe bound on he op imal ac ion ¯
S, as
he o me only conside s a ia ions o he ac ion in he di ec ion o he s aigh pa h [44].
I is hus expec ed (and explici ly shown in [45]) ha his app oxima ion will di e ge om
he ull solu ion as he dimensionali y o he po en ial is inc eased.
We ound ha in his case he dis ibu ion o ac ions in e ms o alse acuum heigh
is iden ical o he op imal one shown in igu e 5, hough sligh ly shi ed o highe alues.
As we can see om igu e 6, he change in he median is minimal when he s aigh -pa h
app oxima ion is conside ed. Al hough, as we jus men ioned, he s aigh -pa h app oxima-
ion is no expec ed o gi e p ecise esul s o po en ials in a highe ield space dimension,
his esul sugges s ha i would be in e es ing o explo e he alidi y o his me hod wi h
GRFs in highe dimensions. Indeed, due o he compu a ional complexi y o such an anal-
– 15 –
JHEP05(2020)142
FV heigh
-2-1 0 1 2 3 4 5
0π/8π/4 3π/8π/2θ
0.0
0.2
0.4
0.6
0.8
1.0
Pθ
Figu e 7. Exi angle dis ibu ion wi h espec o di ec ion o he lowes eigen alue o he ins an on
pa h o he mos p obable decay channel in each gene a ed po en ial.
ysis, a ough s a is ical es ima e o he decay a e ob ained wi h his app oxima ion would
s ill be e y aluable.
4.3 The lowes ac ion
In many ci cums ances one will be in e es ed in he lowes ac ion o a pa icula kind o
minima. This will o cou se co espond o he pa h ha would domina e he decay o
hose minima. In his subsec ion we will in es iga e he cha ac e is ics o such ajec o ies
in ield space.
4.3.1 Exi angle
An in ui i e way o hink abou he mos likely decay p ocess would be o imagine ha
he unneling occu s along he ajec o y wi h he lowes ba ie . One can check his idea
in ou case by i s iden i ying he angle (in ou 2d ield space), θ, ha he ins an on
ajec o y makes wi h espec o he di ec ion o he lowes eigen alue o he Hessian a
he minimum. A dis ibu ion o such angles ob ained o di e en alues o he heigh is
plo ed in igu e 7. We see ha he e is a clea endency o he unnelings o occu a ound
θ≈0 bu he co ela ion is no e y s ong.
4.3.2 Es ima ing he lowes ac ion
The co ela ion o he ins an on pa h wi h he lowes eigen alue di ec ion a he alse
acuum sugges s ha one can y o es ima e he lowes ac ion by analyzing he po en ial
along he lowes eigen alue di ec ion alone. This has been ecen ly p oposed in he con ex
o he landscape in [26]. In he ollowing we will use ou la ge sample o ealiza ions o es
his idea in de ail in ou 2dGRF model o he landscape.
– 16 –
JHEP05(2020)142
◆
◆
◆
◆
◆◆◆◆
Min. ac ion pe po en ial
◆Sa id app. along lowes eig.
-2-1 0 1 2 3 4 5 V(ϕFV)/U0
1
100
103
104
105
S
Figu e 8. Dis ibu ion o lowes ac ion pe po en ial and Sa id app oxima ion [46] along he lowes
ba ie di ec ion, in e ms o alse acuum heigh . The i in eq. (4.6) is shown o compa ison wi h
p e ious esul s.
In o de o e alua e he ins an on ac ion along he lowes eigen alue di ec ion we i s
ake a slice o he po en ial along ha di ec ion and i i o be o he o m,
Vle(φ1) = V0+1
2λ1φ2
1+1
3!ρ111φ3
1+1
4!δφ4
1.(4.8)
No e ha his p ocedu e does no gua an ee ha he esul ing one-dimensional po en-
ial is sui able o a unneling p ocess. In ac , in many cases he po en ial cons uc ed his
way does no ha e a lowe minimum along his di ec ion and he e o e i canno be used
o es ima e he decay a e. In he ollowing we will only compu e he ins an on ac ion in
he success ul cases whe e his 1d unca ion gi es an accep able o m, wha in pa icula
equi es ρ111 <0.
Conside ing his simple o m o he po en ial as he mos likely exi pa h o he
decay ansi ion we can es ima e he ins an on ac ion. In o de o do ha we will use he
pa ame iza ion o he Euclidean ac ion o he bounce ha was ob ained by Sa id in [46].
In ou no a ion his becomes,
¯
SS=


18λ1
ρ111245.4−46.1 + 2π2
12(1−4κ)3+16.5
(1−4κ)2+28
1−4κ, κ > 0
18λ1
ρ111245.41+(136.2
2π2)1.1|κ|1.1−1/1.1, κ ≤0
(4.9)
whe e
κ=3
4δλ1
ρ2
111
.(4.10)
We show in igu e 8 he dis ibu ions o he lowes ac ion om he exac compu a ion
and compa e i o his es ima e along he lowes ba ie di ec ion. We no ice ha he
ag eemen be ween hese wo esul s is p e y good, wha sugges ha one can use his
– 17 –
JHEP05(2020)142
app oxima ion o es ima e he decay a e o acua in a Gaussian andom landscape. Mo e-
o e , i is wo h no ing ha his app oxima ion depends only on he local s uc u e o he
minimum, p ecisely whe e he Slepian model has a la ge p edic i e powe o la ge alues o
V . The exp ession (4.9) becomes inc easingly accu a e o la ge alues o he alse acuum
ene gy V , wha indica es ha in his egime ins an on ac ion is mos ly de e mined by he
local o m o he minimum. On he o he hand, acco ding o he Slepian model, he scala
po en ial a ound all high minima should look e y simila in all ealiza ions, wi h i s shape
domina ed by he i s e m in (3.10). This explains why he dis ibu ion o ins an on
ac ions becomes mo e de e minis ic ( igu e 5) o la ge alues o V , and he e o e also
he ag eemen be ween he Sa id app oxima ion (4.9) o he lowes ac ion and ou i in
eq. (4.6) o he median o he dis ibu ion.
I would be in e es ing o check i his good ag eemen pe sis s on a much la ge
landscape wi h hund eds o di ec ions in ield space,10 and whe he he app oxima ion (4.9)
can be used in combina ion wi h ou Slepian model make obus p edic ions ega ding he
unneling a es o high acua.
5 In la ion in a Slepian andom landscape
Up o now we ha e been using all he so wa e and ma hema ical ools desc ibed abo e o
he compu a ion o bounce p o iles and ac ions wi h Gaussian andom ields condi ioned
o ha e a minimum a φ=0. In his sec ion, we u n o s udying cons ained GRFs wi h
in lec ion poin s a he o igin o ield space ocusing on hei applica ion o cosmological
in la ion.
In la ion in andom po en ials has al eady been ex ensi ely s udied [13,14,19,20,47].
Mo e speci ically, in la ion a ound in lec ion poin s has ecei ed special a en ion o being
capable o sus aining enough e- olds o make con ac wi h obse a ions, while aking place
in a small egion o ield space wi h an e ec i ely one-dimensional po en ial.
While mos o he ob ained esul s and dis ibu ions seem p omising, hey ha e only
been es ed wi hin Taylo expansions a ound hese poin s, ins ead o using ull GRFs.
As we men ioned be o e, such me hods do no cap u e co ec ly he global ea u es o
he po en ial, wha is essen ial o cha ac e ising he non-pe u ba i e s abili y o acua.
The e o e, his p ocedu e is unsui able o s udying models o in la ion whe e he ini ial
condi ions a e de e mined by he decay o a pa en alse acuum.
In his sec ion we will apply Slepian models o cons ain Gaussian andom ields o
ha e in lec ion poin wi h he desi ed p ope ies o sus ain in la ion, and hen we will s udy
he dependence o i s cosmological obse ables on he ini ial condi ions, se by di e en
ealiza ions o he pa en acuum.
10No e ha in ou calcula ion we kep he qua ic e m o he po en ial while in e e ence [26] he
au ho s d op his e m a guing ha o la ge numbe o ields (N) his coe icien becomes i ele an . We
ha e checked ha in ou case his is no he case and in o de o ob ain a good ag eemen i is necessa y o
ake his e m in o accoun . This is due o he ac ha we ha e limi ed ou in es iga ion o he N=2 case.
– 18 –
JHEP05(2020)142
5.1 1D in lec ion poin in la ion
Le us b ie ly e iew he main esul s o one-dimensional in lec ion-poin in la ion
(see [20,48] and e e ences he ein o mo e de ails). Le us conside a po en ial o he o m,
V(φ) = u+ηφ +1
6ρφ3,(5.1)
whe e, in o de o sa is y he slow- oll condi ions a ound he in lec ion poin , we will
assume ha ηu. No e ha we do no need o assume ha he hi d de i a i e is oo
small. In ac , ollowing ypical condi ions o a GRF we will conside he case whe e
uρ. Taking his in o accoun one can show ha slow- oll in la ion condi ions will be
sa is ied in he in e al
−u
ρ<φ<u
ρ,(5.2)
which oge he wi h he condi ion uρimplies ha we a e desc ibing small ield
in la ion. Using he slow- oll condi ions, i is easy o check ha he expec ed numbe o
e- olds, Nexp, ha can be achie ed wi hin ha egion is
Nexp =Zu/ρ
−u/ρ
dφ
√2≈π√2u
√ηρ −4≡Nmax −4,(5.3)
whe e = (V00(φ)/√2V(φ))2and Nmax is he maximal numbe o e- olds achie able in he
whole po en ial. Mo eo e , de ining
x≡πNCMB
Nmax
, y ≡Nmax
2π,(5.4)
whe e NCMB is he e- old numbe a which he CMB scales lea e he ho izon, he spec al
index o scala pe u ba ions can be shown o be gi en by
ns= 1 + 2
y an x−y
1 + y an x.(5.5)
Finally, he ampli ude o scala pe u ba ions can be exp essed as
∆2
R=1
12π2
V3(φ)
V0(φ)2≈N4
CMBρ2
48π2u 2(x, y) (5.6)
whe e
(x, y) = cos2(x)(y an(x) + 1)2
x2(y2+ 1) ,(5.7)
sa is ies (x, y)∼1 o y1 and x∼1.
Wi h hese exp essions a hand, we can easily ob ain a se o pa ame e s o he in lec-
ion poin (u, η and ρ) ha a e in ag eemen wi h he cu en cosmological obse a ions,
namely, Nexp > NCMB ≈50, ns≈0.965 and ∆2
R≈2×10−9(see eq. (5.8) below).
– 19 –

JHEP05(2020)142
5.2 Nume ical in lec ion poin s in a 2D landscape
We now wan o embed 1din lec ion-poin in la ion in ou 2dGRF landscape. In o de
o do ha we can ollow he p ocedu e explained in sec ion 3.2 o Slepian models in he
case o in lec ion poin s. In he no a ion in oduced ea lie , he 1dpa ame e s η=η1and
ρ=ρ111, co espond o he de i a i es along he la di ec ion o he mul idimensional
in lec ion poin . No e ha , in p inciple, uand ρ111 (when e alua ed a he same poin ) a e
unco ela ed, bu he same is no ue o uand he second de i a i e along he in la on
di ec ion λ1; simila ly η1and ρ111 a e also co ela ed, see eq. (3.17). He e we a e in e es ed
in s udying he global p ope ies o he landscape on he cosmological obse ables so we
will ocus on a pa icula ype o in lec ion poin whe e we ha e ixed i s 1dpa ame e s.11
Following he s eps om he p e ious sec ion, we buil wo-dimensional GRFs wi h an
in lec ion poin whose in la ing di ec ion has ixed ea u es. In he o hcoming sec ions
we se
u= 0.5U0, η1= 6.8·10−6U0
Λ, ρ111 = 2.5U0
Λ3(5.8)
whe e U0= 6.0·10−16 M4
Pl and Λ = 0.5MPl de ine he ene gy scale and co ela ion leng h
espec i ely, wi h he Planck masses w i en explici ly o cla i y.
Once u,ηand ρha e been ixed, using he p obabili y dis ibu ions lis ed in (3.16)
and (3.17), we can ob ain he emaining pa ame e s o he wo-dimensional in lec ion poin
se a he o igin o ield space φ=0, and gene a e in a e icien way a la ge sample o
GRFs wi h he lis ed p ope ies.12
As an example, we show in igu e 9a ield cons uc ed wi h he abo e cons ain s. We
hen used AnyBubble o unnel om a highe alse acuum o he cen al in lec ion poin .
We no e ha e en hough in e e y ealiza ion he in lec ion poin has he same p ope ies
along he φ1di ec ion up o hi d o de , he po en ials a e di e en away om ha poin .
This means ha he alse acuum, which decays o he egion a ound he in lec ion poin ,
is loca ed in a di e en place and i also has di e en ea u es in each ealiza ion, e.g.
acuum ene gy and ba ie heigh . Using AnyBubble we compu ed he exi poin s o a
la ge se o ealiza ions. A e ha we used hese exi poin s o he ins an on decay as he
s a ing poin s o a Lo en zian e olu ion o a FRW uni e se wi h his po en ial.
In o de o s udy he in la iona y ajec o y we used mT anspo [49], a Ma hema ica
code de eloped o compu e in la iona y obse ables using he anspo me hod. The
cosmological e olu ion inside o a bubble uni e se c ea ed om unneling is desc ibed by
an open FRW uni e se [50]. He e, o simplici y, we used he la -space app oxima ion o
he e olu ion o he cosmological in e io o he bubble.13
11I is also in e es ing o s udy he e ec s o a ying hese pa ame e s oge he wi h he global p ope ies
o he GRF. We lea e he de ails o his calcula ion o a u u e publica ion.
12No e ha ollowing ou ea lie de ini ion o he in lec ion poin in ou 2dlandscape, we ha e se η2= 0
and λ2>0.
13No e ha in eali y he ini ial cosmological e olu ion is domina ed by he spa ial cu a u e o he open
FRW slices ha desc ibe he bubble in e io . This will ha e some e ec on he ini ial s ages o he e olu ion
o he scala ield in a mul idimensional po en ial. See [19,28] o a discussion o hese e ec s.
– 20 –
JHEP05(2020)142
Figu e 9. A Gaussian andom ield condi ioned o ha e an in lec ion poin in he middle. The
dashed line ep esen s he unneling om a minimum o a lowe in lec ion poin . The in la iona y
slow- oll phase s a s a he exi poin , in la es o a ound 124 e- olds ollowing he solid line, and
e ol es owa ds he closes minimum. We only show he in la iona y pa o he ajec o y. G een,
yellow and ed do s ep esen minima, saddle poin s and maxima o he po en ial. The in lec ion
poin is ma ked wi h a blue do .
In he example om igu e 9, he dashed line ep esen s he unneling ajec o y, while
he solid one ma ks he in la iona y one. We ound his pa h o sus ain a o al o 124.1
e- olds and a spec al index o ns= 0.964 a he obse able scale.
5.3 S a is ics o in la iona y pa ame e s
In o de o es he me hod desc ibed abo e o gene a e in la iona y andom ields, we
gene a ed 5000 GRFs cons ained o ha e an in lec ion poin wi h he same p ope ies as
he one in he example o igu e 9(see eq. (5.8)). Nex , in each o hese ealiza ions, we
ound all minima lying abo e he cen al in lec ion poin and used anyBubble o compu e
he unneling ajec o y om he o me o he la e in each case. Conside ing he exi
poin as he s a ing poin o an in la iona y phase, we used mT anspo o ind he numbe
o e- olds, powe spec um, enso - o-scala a io, spec al index and i s unning. The
dis ibu ions o he e- old numbe and he spec al index a e shown in igu e 10, o a pi o
scale o 50 e- olds, whe eas he ac ion associa ed o he unneling o he in lec ion poin is
shown in igu e 11. This is a di e en dis ibu ion han he ones we ound ea lie , since he
common ac o in hese decays is he inal poin and we do no impose any hing abou he
ini ial ( alse acuum) s a e. I is in e es ing o see ha his dis ibu ion is qui e peaked
a ound an ac ion o he o de o 103.
– 21 –
JHEP05(2020)142
110 120 130 140 150Ne
0.00
0.05
0.10
0.15
0.20
0.25
PNe
(a)
0.960 0.965 0.970 0.975
ns
0
100
200
300
400
Pns
(b)
Figu e 10. (a) Dis ibu ion o numbe o e- olds, wi h Nexp shown wi h a dashed line (b) His og am
o he ob ained spec al index, wi h he analy ic p edic ion ma ked wi h a dashed line. Bo h igu es
ep esen 4000 in la iona y ajec o ies (see ex ).
We ha e also ob ained he dis ibu ions o he ampli ude o scala pe u ba ions,
enso - o-scala a io and unning o spec al index which u ned ou he be cen e ed
a ound he alues
∆2
R= (2.02±0.04)·10−9, = (8.0±0.1)·10−9and α= (−2.49±0.02)·10−3,(5.9)
espec i ely.14 Ou esul s in his sec ion a e ully compa ible wi h he 1ds udies in [14].
Finally, in igu e 12, we show se e al in la iona y ajec o ies co esponding o un-
nelings in di e en GRFs wi h an in lec ion poin in he middle wi h he same ea u es.
No e ha all ajec o ies, no ma e how a hey s a om, ha e a simila beha io .
A e oscilla ing in he e ical φ2di ec ion, hey all s abilize a ound he in lec ion poin
and in la e along i . Mos o he e- olds happen in he icini y o he in lec ion poin , as
p edic ed by he analy ic es ima ion.
We ha e ob ained success ul esul s om his analysis a ound 80% o he imes. The
es o he imes he p ocedu e did no yield a cosmological solu ion in ag eemen wi h
ou uni e se ei he because in la ion ended oo soon o because he exi poin was oo a
om he cen al in lec ion poin and he in la on ajec o y wen as ay. The success ul
pa hs show e y good ag eemen wi h he 1d esul s p esen ed in he p e ious sec ion. We
see ha e en hough some o he ajec o ies ha e some subs an ial de ia ion om he 1d
in la iona y di ec ion, he cosmological obse ables a e s ill in p e y good ag eemen wi h
he single ield in lec ion poin in la ion. The dis ibu ions o he esul s a e qui e peaked
a ound hei cen al alues, so we can conclude ha he dependence o he obse ables on
he ini ial condi ions seems o be qui e mild.
14The cosmological e olu ion o hese Lo en zian ajec o ies con inue a e in la ion un il hey each
a lowe minimum. We ha e no ine- uned his minimum o be in Minkowski space, so in gene al he
e olu ion leads o e e nal de Si e o o an An i-deSi e c unch. We a e only in e es ed in he s a is ics
o he in la iona y pe iod so we ha e s opped his e olu ion a e he ield lea es he slow- oll egime.
– 22 –
JHEP05(2020)142
2 4 6 8 Log10[S]
0.0
0.1
0.2
0.3
0.4
PS
_
Figu e 11. Dis ibu ion o he unneling ac ion om a minimum o he cen al in lec ion poin ,
igh be o e in la ion begins.
0.05 0.10 0.15 0.20
ϕ1
-0.2
-0.1
0.1
ϕ2
Figu e 12. Showcase o se e al in la iona y ajec o ies om di e en unnelings o he cen al
in lec ion poin . Each exi poin is ma ked by a blue do .
I is impo an o emembe ha all hese ealiza ions ha e he same 1din lec ion poin
pa ame e s. In o de o ex ac he comple e s a is ical in o ma ion abou he p edic ions
o a pa icula GRF we should combine hese esul s wi h he ones ob ained om in lec ion
poin s wi h o he pa ame e s wi h hei co ec s a is ical weigh . This is a much mo e
nume ically in ensi e p oblem and we lea e i o a u u e publica ion.
– 23 –
JHEP05(2020)142
Le us change he no a ion o ∂φjV(0) = V0
j(0) and e alua e he p e ious exp ession o
some use ul cases:
hV(0)V(0)i=U2
0(A.25)
hV(0)V0
i(0)i=V0
i(0)V00
jk(0)= 0 (A.26)
V0
i(0)V0
j(0)=−V(0)V00
ij (0)=−∂2C(0)
∂φi∂φj
=α2δij (A.27)
V00
ij (0)V00
kl(0)=∂4C(0)
∂φi∂φj∂φk∂φl
=


α22 i i=j6=k=l(and pe ms.)
α4i i=j=k=l
0 o he wise.
(A.28)
V(0)V000
jkl(0)=V00
ij (0)V000
klm(0)= 0 (A.29)
V0
i(0)V000
jkl(0)=−V00
ij (0)V00
kl(0)=


−α22 i i=j6=k=l(and pe ms.)
−α4i i=j=k=l
0 o he wise.
(A.30)
V000
ijk(0)V000
lmn(0)=−∂6C(0)
∂φi∂φj∂φk∂φl∂φm∂φn
=






α222 i i=j6=k=l6=m=n(and pe ms.)
α24 i i=j6=k=l=m=n(and pe ms.)
α6i i=j=k=l=m=n
0 o he wise.
(A.31)
In he abo e exp essions, αi,αij and αijk a e nume ical cons an s which depend only on
he co a iance unc ion o he (uncons ained) Gaussian andom ield. No e ha in he
wo-dimensional case α222 will be absen om all de i a ions, since he indices appea ing
in he co ela ion unc ion be ween he hi d de i a i es can only ake wo di e en alues.
No e also ha odd de i a i es o he GRF a e unco ela ed wi h e en ones when hey
a e e alua ed a he same poin in ield space. This is due o he iso opy o he co a iance
unc ion: i i is w i en as a powe se ies, only e en powe s such as φ2
i, φ2
iφ2
jwill be
in ol ed. The e o e, only hose co ela ions which end up in ol ing e en de i a i es o he
co a iance unc ion a e non-ze o.
This howe e , does no mean he ields V(φ) and, say, V0
i(φ) a e comple ely unco e-
la ed. I we e alua e hem a di e en poin s in ield space, i can be shown [30, heo em 2.3]
ha
V(φ)V0
i(0)=−∂
∂φi
C(φ) (A.32)
V(φ)V00
ij (0)=∂2
∂φi∂φj
C(φ) (A.33)
V(φ)V000
ijk(0)=−∂3
∂φi∂φj∂φk
C(φ) (A.34)
he e o e, a GRF and any o i s de i a i es a e co ela ed as p ocesses.
– 30 –

JHEP05(2020)142
A.5 The Kac-Rice o mula and condi ioned Gaussian andom ields
Conside a Gaussian andom ec o ield wi h componen s V(φ) = {V1(φ), . . . , Vn(φ)}.
The mul idimensional16 Kac-Rice o mula o his ield gi es us he expec ed numbe o
imes a ce ain e en , say, V(φ) = u, happens in an in e al φ∈Io olume V:
E#,I [V(φ) = u] = ZI
dφ|de V0(φ)|δ(V(φ)−u)(A.35)
whe e de V0(φ) s ands o he Jacobian de e minan o he ec o ield,17 ha is,
V0(φ) = 


∂φ1V1(φ)··· ∂φ1Vn(φ)
.
.
..
.
.
∂φnV1(φ)··· ∂φnVn(φ)


.(A.36)
I he ield is s a iona y, ha is, homogeneous and iso opic, we can simpli y he exp ession
abo e. Deno ing V0=V(0) and V0
0=V0(0), we ind, assuming e godici y,
E#,I [V(φ) = u] = VZdV0dV00|de V00|δ(V0−u)P(V0,V00) (A.37)
whe e he in eg al is pe o med o e he whole domain o V0and V0
0and P(V0,V00) is he
join PDF o V0and i s de i a i es.
Mo e han one simul aneous e en can be conside ed in he exp essions abo e by
enla ging he ec o Vand in oducing mo e Di ac del as ep esen ing each e en .18
While he abo e exp ession can ce ainly be used o ob ain he numbe o imes a
ce ain e en happens in a gi en in e al, i can also be used o ob ain dis ibu ion unc ions.
Mo e speci ically, applying e godici y heo ems, i can be shown [30] ha he p obabili y
o an e en Ahappening, gi en ha Bhas happened, ha is, P(A|B), can be ob ained by
P(A|B) = E#,I [A∩B]
E#,I [B].(A.38)
I Adepends on con inuous pa ame e s (such as he posi ion in ield space o he GRF),
hen he exp ession abo e ep esen s a p obabili y dis ibu ion unc ion.
A.6 Condi ioned Gaussian andom ield o a c i ical poin
Wi h he ools p esen ed in he sec ions abo e, we a e now eady o begin condi ioning
GRFs. We can begin applying (A.38) and specializing i o c i ical poin s. We deno e
by A he e en desc ibing he ield V(φ) aking a pa icula con igu a ion, while Bim-
poses V(0)≡V0=uand V0
i(0)≡ηi= 0, ha is, a c i ical poin lying in he cen e
o ield space a heigh u. In o de o p oceed mo e easily, we shall disc e ize V(φ) as
{V(φ1), . . . , V (φq)}≡{V1,...Vq} ≡ V.
16No e ha his o mula is only alid o ields mapping Rn→Rn.
17Fo c i ical poin s, he Jacobian is iden ical o he Hessian o he GRF a he c i ical poin .
18See, howe e , [30, ch.8] o a discussion on di e en ypes o condi ioning e en s and how o deal wi h
hem. The eason why we conside he V0=ue en simply wi h a Di ac del a is ha i is a e ical window
condi ioning e en .
– 31 –
JHEP05(2020)142
In his case, he condi ioning e en in ol es he Gaussian andom ec o ield V=∇V,
whose Jacobian is he Hessian o he o iginal ield Ve alua ed a φ=0. The e o e, i s
de e minan is simply he p oduc o he eigen alues o he Hessian e alua ed a he o igin,
Qn
i=1 λi.
Applying he Kac-Rice o mula (A.37) in o (A.38) yields
PV(φ)V0=u,∇V0=0≡Pcp[V(φ)] =
=Zn
Y
i=1 dηiδ(ηi)dλi|λi|∆(λ)δ(V0−u)
q
Y
j=1 d˜
Vjδ(˜
Vj−Vj)PV0,V,η,λ
Zn
Y
i=1 dηiδ(ηi)dλi|λi|∆(λ)δ(V0−u)PV0,η,λ(A.39)
=NZn
Y
i=1
(dλi|λi|)∆(λ)PV(φ),λ1,...,λnV0=u, ∇V0=0(A.40)
whe e he in eg a ion domain will depend on he kind o c i ical poin we a e wo king
wi h. ∆(λ)∝Qi<j |λi−λj|is he Jacobian o he a iable change om componen s
o he Hessian ma ix o i s eigen alues, he p opo ionali y cons an depending on he
dimensionali y o he ield space. Fo simplici y, he denomina o in (A.39) has been
conside ed as a no maliza ion ac o o he dis ibu ion in he nume a o .
We can ew i e (A.39) in a mo e use ul way:
Pcp[V(φ)] = Y
iZdλiqu(λ1, . . . , λn)PV( )V0=u, ∇V0=0, λ1, . . . , λn(A.41)
whe e
qu(λ1, . . . , λn) = Y
i|λi|∆(λ)Pλ1, . . . , λnV0=u, ∇V0=0(A.42)
ep esen s he dis ibu ion o he Hessian eigen alues a he o igin o a c i ical poin o
heigh u. Howe e , due o he homogeneous and iso opic na u e o he o iginal GRF, he
la e dis ibu ion is alid o any c i ical poin in he GRF, hus gi ing us a dis ibu ion
o he pa ame e s a c i ical poin s in he uncons ained ield.
Equa ions (A.41) and (A.42) a e cen al esul s in his de i a ion. No e ha he
Qi|λi|∆(λ) ac o is a di ec consequence o he Kac-Rice o mula, and as we shall ex-
plici ly see in appendix B, i ca ies impo an consequences in he dis ibu ion o he
eigen alues a c i ical poin s.
We can now see he powe o his me hod. Assuming we ha e disc e ized ou ield space,
we can eadily compu e he condi ional p obabili y dis ibu ions in (A.41) and (A.42) using
he esul s om sec ion A.2. This leads, oge he wi h (A.42), o a dis ibu ion om which
we can d aw eigen alues o a minimum o heigh u. These can be plugged in (A.41) o
gene a e i e a ions o GRFs wi h a minimum (o any o he c i ical poin ) a hei o igin.
In o de o apply all his machine y, le us in oduce he ollowing Gaussian andom
ec o :
{V(φ1), . . . , V (φq), V (0), V 0
1(0), . . . , V 0
n(0), V 00
11(0), . . . , V 00
nn(0), V 00
12(0), . . . , V 00
(n−1)n(0)
| {z }
V00
ij (0)i<j
}
(A.43)
– 32 –
JHEP05(2020)142
whe e we deno e by φq he posi ion in ield space o a disc e e se o poin s whose cen e is
loca ed a 0,V0
i(0) desc ibes he i s de i a i e along φiand V00
ij (0) is he (i, j)- h elemen
o he Hessian ma ix. In o de o unclu e he no a ion, we will compac i y he p e ious
ec o as
{V, V (0),V0(0),V00(0)}(A.44)
which has dimension q+1+n+n+1
2n(n−1). The mean o (A.43) is ze o, and he
co a iance ma ix o hese quan i ies can be compu ed om he esul s in sec ion A.4:
Σ = 




SV V SV0SV1SV2
S0VU2
00S02
S1V0S11 0
S2VS20 0S22





(A.45)
whe e
S02 =−α2··· −α20··· 0=ST
20 (A.46)
S11 =α2×1n(A.47)
S22 =













α4α22 ··· α22
α22 α4··· α22 0
.
.
..
.
.....
.
.
α22 α22 ··· α4
α22 0
0...
0α22













(A.48)
SV V =





C(0)C(φ1−φ2)··· C(φ1−φq)
C(φ2−φ1)C(0)··· C(φ2−φq)
.
.
..
.
.....
.
.
C(φq−φ1)C(φq−φ2)··· C(0)






(A.49)
S0V=C(φ1)C(φ2)··· C(φq)=ST
V0(A.50)
S1V=





−C0
1(φ1)−C0
1(φ2)··· −C0
1(φq)
−C0
2(φ1)−C0
2(φ2)··· −C0
2(φq)
.
.
..
.
.....
.
.
−C0
n(φ1)−C0
n(φ2)··· −C0
n(φq)






=ST
V1(A.51)
S2V=











C00
11(φ1)··· C00
11(φq)
.
.
.....
.
.
C00
nn(φ1)··· C00
nn(φq)
C00
12(φ1)··· C00
12(φq)
.
.
.....
.
.
C00
(n−1)n(φ1)··· C00
(n−1)n(φq)











=ST
V2(A.52)
– 33 –
JHEP05(2020)142
In o de o simpli y he no a ion, since he join ly Gaussian p obabili y dis ibu ion in
he end depends on wo-poin unc ions, we can ac ually w i e19 (A.45) in he ollowing way:
Σ = 






U2
0C(φ1−φ2)C(φ1)SV1(φ1)SV2(φ1)
C(φ2−φ1)U2
0C(φ2)SV1(φ2)SV2(φ2)
C(φ1)C(φ2)U2
00S02
S1V(φ1)S1V(φ2)0S11 0
S2V(φ1)S2V(φ2)S20 0S22







(A.53)
whe e
SV1(φ) = −C0
1(φ)··· −C0
n(φ)=ST
1V(A.54)
SV2(φ) = C00
11(φ)··· C00
nn(φ)C00
12(φ)··· C00
(n−1)n(φ)=ST
2V(A.55)
Wi h hese a angemen s, he Gaussian andom ec o co esponding o (A.53) is
V(φ1), V (φ2), V (0),V0(0),V00(0).(A.56)
We ha e decomposed (A.53) in o blocks so i can be plugged in o (A.57) and (A.58)
o ob ain he mean unc ion and co a iance ma ix o he condi ioned p ocess.20 Using
he esul s gi en abo e, one ge s ha he expec a ion alue o he GRF a ound a c i ical
poin whe e V0=uand V0
0= 0, is gi en by,
˜µ(φ) = µ(φ) + C(φ)SV1(φ)SV2(φ)


U2
00S02
0S11 0
S20 0S22



−1


u
0
h



=C(φ)SV2(φ) U2
0S02
S20 S22 !−1 u
h!(A.57)
whe e h=h11, . . . , hnn, h12, . . . , h(n−1)n ep esen s a ce ain con igu a ion o he Hessian
componen s o he ield a ound he o igin.
Fu he mo e, he co a iance unc ion o he condi ioned GRF is now
˜
C(φ1,φ2) = C(φ1−φ2)−C(φ1)SV1(φ1)SV2(φ1)


U2
00S02
0S11 0
S20 0S22



−1


C(φ2)
S1V(φ2)
S2V(φ2)


=C(φ1−φ2)−C(φ1)SV2(φ1) U2
0S02
S20 S22 !−1 C(s)
S2V(φ2)!
−SV1(φ1)S−1
11 S1V(φ2) (A.58)
19We basically ha e e alua ed he i s ow o a gi en φ1and he i s column o a gi en φ2, jus as
in [32]. Doing so allows us o ea he independen a iable as a con inuous one, a he han a disc e e one.
20S ic ly speaking, we should be ge ing he mean and co a iance o he andom ec o {V(φ1), V (φ2)}.
Due o he iso opy o he GRF, φ1and φ2can be any poin s in ield space. Thus, in o de o unclu e
he no a ion, we will only keep ack o a single componen o he esul ing mean ec o . Likewise, we will
only keep he hV(φ1)V(φ2)icomponen o he co a iance ma ix.
– 34 –
JHEP05(2020)142
We can also ob ain (A.42), he dis ibu ion o eigen alues a a c i ical poin o a gi en
heigh u, ollowing he same s eps as abo e, using as ini ial co a iance ma ix he bo om-
igh block o (A.53).
A.6.1 Analysis o a condi ioned 2D Gaussian ield
Le us apply hese exp essions o a wo-dimensional iso opic and homogeneous GRF wi h
co a iance unc ion
C(φ) = U2
0exp −φ2
2Λ2,(A.59)
and ze o mean. Fo his case, we ob ain he condi ioned mean om (A.57), which gi es
˜µ(φ) = e−φ2
2Λ2"u1 + φ2
2Λ2+1
2φ1φ2 h11 h12
h21 h22 ! φ1
φ2!#,(A.60)
whe e h21 =h12, by de ini ion. Since we a e ee o choose he basis o φ, in o de
o simpli y he exp ession we will employ he eigen ec o basis o he Hessian ma ix,
he e o e ans o ming (A.60) o
˜µ(φ) = e−φ2
2Λ2"u1 + φ2
2Λ2+1
2
2
X
i=1
λiφ2
i#,(A.61)
whe e λideno e he wo eigen ec o s, d awn om (A.42) specialized o his case (see
below). As o he condi ioned co a iance, om (A.58) we ob ain
˜
C(φ1,φ2) = U2
0exp−|φ1|2+|φ2|2
2Λ2expφ1·φ2
Λ2−1−φ1·φ2
Λ2−(φ1·φ2)2
2Λ4.(A.62)
No e ha he co a iance unc ion o he condi ioned p ocess is no homogeneous anymo e!
This, howe e , makes comple e sense. We ha e ac ually made he cen e o e e y ealiza ion
special, meaning ha homogenei y is b oken in his sense. In ac , he new co a iance is
iso opic wi h espec o φ=0, u he s a ing ha he cen e o he GRF is somehow
di e en om he es o he poin s.
All he p esen ed machine y wo ks no only o minima, bu also o maxima and
saddle poin s as well; he only di e ence among hese being he sign o each λi.
A.6.2 Dis ibu ion o heigh s and eigen alues o he Hessian a a c i ical poin
In o de o calcula e he p obabili y dis ibu ion o he eigen alues o he Hessian a a
ce ain heigh o he po en ial a c i ical poin s we should pay a en ion o wo ing edien s.
The i s one is he ac ha he heigh and he second de i a i es a e co ela ed, so
we need o calcula e he mul i a ia e co a iance unc ion o hese quan i ies oge he .
Fu he mo e, we also wan o calcula e his a c i ical poin s which can be done wi h he
use o he gene alized Kac-Rice o mula.
Assuming a c i ical poin loca ed a φ=0, he p obabili y dis ibu ion o be com-
pu ed is
PV0, λ1, λ2∇V0=0(A.63)
– 35 –

JHEP05(2020)142
We can easily compu e he PDF by condi ioning he ollowing andom ec o :
{V0, h11, h22, h12, η1, η2}(A.64)
o mean ze o and co a iance ma ix



U2
0S02 0
S20 S22 0
0 0 S11


(A.65)
Applying (A.19) and (A.20) o ob ain he mean and co a iance o he condi ioned
p ocess and plugging hem in o (A.42), we ge
Pcp(V0, λ1, λ2)du
2
Y
i=1
dλi=N |λ1||λ2|∆(λ)PV0, λ1, λ2∇V0=0(A.66)
=N|λ1−λ2||λ1||λ2|exp −V2
0
2U2
0exp "−Λ2λi+V0
2U02#dλidV0
(A.67)
whe e Nis a no maliza ion ac o and, in his wo-dimensional example, ∆(λ)=|λ1−λ2|·π/2.
Se ing V0 o a cons an alue, say V0=u, in (A.67) yields he dis ibu ion qu(λ1, λ2),
de ined in (A.42). On he o he hand, in eg a ing ou ei he V0o he eigen alues, gi es
he ma ginal dis ibu ion o he emaining a iables in c i ical poin s (see appendix B o
mo e de ail).
Ano he in e es ing applica ion o (A.66) is ha i can be used o coun he expec ed
numbe o c i ical poin s in a ce ain egion o ield space. Fo example, o compu e he
expec ed numbe o minima pe co ela ion olume Λ2in he example abo e, a di ec
applica ion o (A.37) yields
E(#min)
Λ2=Z+∞
−∞
du Z+∞
0
dλ1Z+∞
0
dλ2
π
2λ1λ2|λ1−λ2|PV0, λ1, λ2∇V0=0
=1
2√3.(A.68)
In his case, he eigen alues ha e been assumed o be posi i e. Se ing o he in eg a ion
limi s can gi e he expec ed numbe o maxima and saddle poin s, o example.
A.7 Condi ioned Gaussian andom ield o an in lec ion poin
We shall de ine an in lec ion poin on ou GRF as a poin whe e he g adien o he ield
poin s in he di ec ion o a Hessian eigen ec o whose co esponding eigen alue is ze o.
Fu he mo e, we will also demand ha he non-ze o eigen alue o he Hessian o be posi i e
a his poin .
In o de o do his we can expand he discussion o he p e ious sec ion by aking
in o accoun he hi d de i a i es o he GRFs along wi h he lowe ones. In o de o
simpli y his desc ip ion we will gi e a de ail accoun o his cons uc ion o a 2dGRF
– 36 –
JHEP05(2020)142
only. Ex ending his o highe dimensions is s aigh o wa d. In pa icula we will be
in e es ed in he Gaussian andom ec o
V(φ1), V (φ2), V0, V 0
1(0), V 0
2(0), V 00
11(0), V 00
22(0), V 00
12(0), V 000
111(0), V 000
122(0), V 000
222(0), V 000
112(0)
(A.69)
whose componen s ha e ze o mean. As o he co a iance ma ix, i can be exp essed as
Σ =









U2
0C(φ1−φ2)C(φ1)SV1(φ1)SV2(φ1)SV3(φ1)
C(φ2−φ1)U2
0C(φ2)SV1(φ2)SV2(φ2)SV3(φ2)
C(φ1)C(φ2)U2
00S02 0
S1V(φ1)S1V(φ2)0S11 0S13
S2V(φ1)S2V(φ2)S20 0S22 0
S3V(φ1)S3V(φ2)0S31 0S33









(A.70)
whe e ( o he 2D case)
SV3(φ) = −C0
111(φ)−C0
122(φ)−C0
222(φ)−C0
112(φ)=ST
3V(A.71)
S13 = −α4−α22 0 0
0 0 −α4−α22 !=ST
31 (A.72)
S33 =




α6α24 0 0
α24 α24 0 0
0 0 α6α24
0 0 α24 α24





(A.73)
and he o he ma ix blocks ha e been de ined in (A.46)–(A.52).
Following he same s eps as in he c i ical poin case, we can ob ain ( o he co a iance
unc ion (A.59)) he exp ession o a GRF once we condi ioned e e y hing up o he hi d
de i a i e. In o de o do his we can i s compu e he mean alue o he GRF in he
icini y o ou in lec ion poin , which is gi en by
˜µ(φ) = 0+C(φ)SV1(φ)SV2(φ)SV3(φ)




U2
00S02 0
0S11 0S13
S20 0S22 0
0S31 0S33





−1




u
η
h
ρ





(A.74)
=C(φ)SV2(φ) U2
0S02
S20 S22 !−1 u
h!+SV1(φ)SV3(φ) S11 S13
S31 S33 !−1 η
ρ!
= exp−φ2
2Λ2
(u+φ·η)1+ φ2
2Λ2+1
2
2
X
i=1
λiφ2
i+1
6
2
X
i,j,k=1
φiφjφkρijk
,(A.75)
whe e he basis o φhas been chosen o be he eigenbasis o he Hessian ma ix (whose
componen s a e desc ibed by hand i s eigen alues by λi) and we ha e deno ed by ηand
ρ he componen s o he i s and hi d de i a i es a he o igin along he eigenbasis.
– 37 –
JHEP05(2020)142
The condi ioned co a iance, on he o he hand, eads
˜
C(φ1,φ2) = C(φ1−φ2)−C(φ1)SV1(φ1)SV2(φ1)SV3(φ1)



U2
00S02 0
0S11 0S13
S20 0S22 0
0S31 0S33




−1



C(φ2)
S1V(φ2)
S2V(φ2)
S3V(φ2)




=U2
0exp−|φ1|2+|φ2|2
2Λ2expφ1·φ2
Λ2−1−φ1·φ2
Λ2−(φ1·φ2)2
2Λ4−(φ1·φ2)3
6Λ6(A.76)
which, once again, is iso opic a ound he o igin o he ield.
A.7.1 P obabili y dis ibu ion o he in lec ion poin pa ame e s
We can ex end he ea men o he eigen alues o he hessian ha we did o he c i ical
poin s o in lec ion poin s. The di e ence is ha we will now impose ha one o he
eigen alues anishes while he o he one is posi i e. Fu he mo e we will also impose
ha he g adien in he second eigen alue di ec ion also anishes. These condi ions ha e
o be included in he calcula ion o he PDF o he pa ame e s o he in lec ion poin s
(V0, η1, λ2,ρ). Using a gene alized e sion o he Kac-Rice p ocedu e we a i e o,
Pin dV0dλ2dη1dρ=N|λ2|2|ρ111|PV0, λ2|λ1= 0 P(η1, ρijk |η2= 0) (A.77)
whe e
PV0, λ2|λ1= 0 dV0dλ2=Nexp −4V2
0−2Λ2V0λ2−Λ4λ2
2
2U0dV0dλ2(A.78)
P(η1, ρijk |η2= 0) dη1dρijk =
Nexp 
−Λ2
12U2
0
18η2
1+ 6Λ2η1(ρ111 +ρ122)+Λ4
2
X
i,j,k=1
ρ2
ijk

dη1dρijk (A.79)
In (A.77), one o he |λ2| ac o s comes om he Jacobian o he a iable change o he
eigenbasis o he Hessian ( hough wi h λ1= 0); he emaining |λ2||ρ111| ac o is jus he
de e minan appea ing in Kac-Rice’s exp ession.
These las exp essions can be used as in (A.68) o compu e he expec ed numbe
o in lec ion poin pe co ela ion olume Λ2, which yields, o ou choice o co a iance
unc ion,
E(#ip)
Λ2=√5−√3
3π.(A.80)
B Nume ical implemen a ion and es s o he p obabili y dis ibu ions
B.1 Gene a ion o Gaussian andom ields: Ka hunen-Lo`e e expansion
In o de o gene a e ealiza ions o wo-dimensional Gaussian andom ields, we eso ed
o he so-called spec al o Ka hunen-Lo`e e decomposi ion, due o i s ma hema ical and
compu a ional simplici y.
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JHEP05(2020)142
Gi en a ce ain mean unc ion µ( ), co a iance unc ion C( ,s) and a disc e ized space
{ a}(whe e a uns o e all npoin s in he la ice space) o a GRF, we can build he ma ix
Cab =C( a, b), which by cons uc ion is symme ic and posi i e de ini e; he e o e, we
can always decompose Cab as
C=UΛUT(B.1)
whe e Λ = diag(λ1, . . . , λn) is he diagonal eigen alue ma ix, consis ing o non-nega i e
en ies, and Uis cons uc ed by inse ing all eigen ec o s along i s ows. Since Λ >0, we
can u he decompose Cas
C=U√Λ√ΛUT=U√ΛU√ΛT=L LT.(B.2)
This p ocedu e is an amoun o pe o ming a Cholesky decomposi ion [56] on C; which is
by a he mos expensi e s ep in his algo i hm, in e ms o compu a ional cos .
Once we ha e compu ed L, cons uc ing he GRF on he disc e ized space is s aigh -
o wa d. We only need o cons uc a andom ec o ξo leng h nwhose en ies a e
independen ly dis ibu ed as Gaussian a iables o ze o mean and uni a iance, and in o-
duce he ollowing a iables:
Va=µa+Labξb,(B.3)
whe e µa=µ( a). I can be easily shown ha his gi es he co ec co ela ions among
he alues o he GRF e alua ed a di e en poin s a,
h(Va−µa)(Vb−µb)i=hLacξcLbdξdi=LacLbdhξcξdi
=LacLbdδcd =LacLbc =LacLT
cb = (LLT)ab =Cab =C( a, b).(B.4)
The main ad an age o using his p ocedu e o gene a e GRFs is ha he main compu-
a ionally cos ly s ep, cons uc ing he Lma ix, needs o be pe o med only once. The es
o he algo i hm is highly i ial om his pe spec i e and allows o u he simpli ica ion,
as we ha e seen.
B.2 Nume ical e alua ions o c i ical poin s
Using he exp essions abo e we can compu e he no malized dis ibu ion o heigh s o
minima, maxima and saddle poin s o a 2dGRF,
Pu,mindu =√3
4πU0e−u2/U2
0−2u
U0+2√πeu2/4U2
0e chu
2U0i+√2πu2
U2
0−1eu2/2U2
0e chu
√2U0idu
Pu,maxdu =√3
4πU0e−u2/U2
02u
U0+2√πeu2/4U2
0e ch−u
2U0i+√2πu2
U2
0−1eu2/2U2
0e ch−u
√2U0idu
Pu,sp du =√3
2√πU0exp−3u2
4U2
0.(B.5)
– 39 –