Ann. Inst. Henri Poincaré Comb. Phys. Interact. 5 (2018), 437–466
DOI 10.4171/AIHPD/59
Moments of quantum Lévy areas
using sticky shuffle Hopf algebras
Robin Hudson, Uwe Schauz, and Yue Wu
Abstract. We study a family of quantum analogs of Lévy’s stochastic area for planar
Brownian motion depending on a variance parameter 1which deform to the classical
Lévy area as ! 1. They are defined as second rank iterated stochastic integrals
against the components of planar Brownian motion, which are one-dimensional Brownian
motions satisfying Heisenberg-type commutation relations. Such iterated integrals can be
multiplied using the sticky shuffle product determined by the underlying Itô algebra of
stochastic differentials. We use the corresponding Hopf algebra structure to evaluate the
moments of the quantum Lévy areas and study how they deform to their classical values,
which are well known to be given essentially by the Euler numbers, in the infinite variance
limit.
Mathematics Subject Classification (2010). 81S25, 46L53.
Keywords. Lévy area, non-Fock quantum stochastic calculus, moments, sticky shuffles,
Euler numbers.
Contents
1 Introduction................................ 438
2 The sticky shuffle product Hopf algebra . . . . . . . . . . . . . . . . . 441
3 Moments and sticky shuffles . . . . . . . . . . . . . . . . . . . . . . . 445
4 Some background about Eulerian and Euler numbers . . . . . . . . . . 451
5 Moments of quantum Lévy areas . . . . . . . . . . . . . . . . . . . . . 458
6 Theclassicallimit............................. 464
References................................... 465
438 R. Hudson, U. Schauz, and Y. Wu
1. Introduction
Lévy’s stochastic area for planar Brownian motion is important in several areas of
modern mathematics and probability theory, ranging from harmonic analysis on
the Heisenberg group to rough noise analysis.
Let us first review the definition of Lévy area as a stochastic integral [13].
Intuitively it is the signed area between the chord joining two time points on
the planar Brownian path and the trajectory between those points. To make this
rigorous, let there be given a planar Brownian motion Band write BD.X; Y / in
terms of components Xand Ywhich are independent one-dimensional Brownian
motions. Let two real numbers a < b be given.
Definition 1. The Lévy area of Bover the time interval Œa; b/ is the stochastic
integral
AŒa;b/ D1
2
b
Z
a
..X X.a//dY .Y Y.a//dX/:
In this definition the integral takes the same value whether it is regarded as of
Stratonovich or Itô type. In the remainder of this paper, however, all stochastic
integrals will be of Itô type, in contrast to [11] where the Stratonovich integral is
used. The latter cannot be defined coherently in a quantum context.
Lévy’s area has interesting connections with classical mathematics through its
characteristic function, which is given by the following theorem.
Theorem 2 (Lévy [12]). We have
EŒexp.izAŒa;b// Dsech .1
2.b a/z/:
We can expand the right-hand side of the formula in Theorem 2using the
Taylor series
sech.z/ D
1
X
mD0
.1/mA2m
.2m/Šz2m ;(1)
where the even Euler zigzag numbers A2m are related to the Riemann zeta function
by
.2m/ D2m
.2m/ŠA2m:(2)
Levin and Wildon in [11] used iterated integrals and combinatorial arguments
arising from the formalism of rough noise to evaluate the moments of AŒ0;1/;which
is tantamount to proving Theorem 2.
Moments of quantum Lévy areas 439
A one-parameter family of quantum Lévy areas has been introduced recently
[7,3]. In these the component one-dimensional Brownian motions of the clas-
sical Lévy area are replaced by a pair of self adjoint operator-valued processes
.P ./.t/; Q./.t//t0:Each such pair is determined by a variance parameter
taking a value in the range 1 < 1:Each of P./ and Q./ is individually a
one-dimensional Brownian motion of variance 2;so that for example, for each
positive time t; P ./.t/ is a Gaussian random variable of mean 0and variance 2t.
But the processes P./ and Q./ do not commute with each other; instead they
satisfy the Heisenberg type commutation relation
ŒP ./.s/; Q./.t/ D 2i min ¹s; tº(3)
in the rigorous Weyl sense that for arbitrary real xand yand nonnegative sand t;
eixP ./.s/eiyQ./.t/ De2ixy min¹s;tºeiyQ./.t/eixP ./.s/ (4)
as unitary operators. Despite their mutual noncommutativity P./ and Q./ can
be regarded as stochastically independent in a certain sense, and hence as the
two components of a quantum planar Brownian motion. Indeed,for arbitrary
real x1; x2; : : : ; xm; y1; y2; : : : ; ynand nonnegative s1; s2; : : : ; sm; t1; t2; : : : ; tn;
the operator PxjP./.sj/CPykQ./.tk/defined on the intersection of do-
mains is essentially self-adjoint, so that the quantum probabilistic expectation
EŒei.PxjP./.sj/CPykQ./.tk//, which in effect determines the joint characteristic
function, is well defined. Moreover this factorizes:
EŒei.PxjP./.sj/CPykQ./.tk//
DEŒeiPxjP./.sj/EŒeiPykQ./.tk/
(5)
and in classical probability such factorization is sufficient for independence.
We use the standard quantum stochastic calculus of [15] in the case when
D1and the non-Fock finite temperature calculus of [8] when > 1 to define
the corresponding quantum Lévy areas, in which the planar Brownian motion is
replaced by its quantum version R./ D.P ./; Q.//:
Definition 3. The quantum Lévy area B./
Œa;b/ of R./ of variance over the time
interval Œa; b/ is the quantum stochastic integral
B./
Œa;b/ D1
2
b
Z
a
..P ./ P./.a//dQ./ .Q./ Q./.a//dP .//:
440 R. Hudson, U. Schauz, and Y. Wu
When D1; the distribution at all times of the corresponding Lévy areas is
degenerate at 0and all moments are zero [3]. For values > 1 the processes R./
generate Type III factorial von Neumann algebras,1whose mutual strong unitary
inequivalence as varies can be regarded as a quantum version of the mutual
singularity of the measures obtained by dilatation of planar Wiener measure
through different dilatation factors :
In view of (3), the normalised standard unit variance Brownian motions
y
P./ D1P./,y
Q./ D1Q./;(6)
become mutually commutative in the limit of large ; so that the corresponding
quantum Lévy areas
y
B./
Œa;b/ D2B./
Œa;b/ (7)
interpolate between the degenerate distribution at D1and the classical case
AŒa;b/ at 1:Thus it is a natural question to ask how the moments behave under
this interpolation and in particular how the Euler zigzag numbers are approached
at 1. The main purpose of this paper is to address this question.
Our method is based firstly on the observation that Definition 1, Definition 3,
and the normalized form of the latter, can be regarded as iterated stochastic
integrals:
AŒa;b/ D1
2Z
a<x<y<b
.dX.x/dY.y/ dY.x/dX.y//; (8)
B./
Œa;b/ D1
2Z
a<x<y<b
.dP ./.x/dQ./.y/ dQ./.x/dP ./.y//; (9)
y
B./
Œa;b/ D1
2Z
a<x<y<b
.d y
P./.x/d y
Q./.y/ dy
Q./.x/d y
P./.y//: (10)
We may thus evaluate moments as expectations of powers, using the so-called
sticky shuffle [6] or stuffle [5] Hopf algebra. Multiplication in this algebra can
be used to express the product of two iterated Itô stochastic integrals as a linear
combination of such iterated integrals. Since the expectation of an iterated integral
vanishes unless each of the individual integrators is time, the recovery formula
[6,1], involving higher order Hopf algebra coproducts, reduces the evaluation of
the moments to a combinatorial counting problem.
1More precisely, it is the the unitary operators eixP ./ .s/ and eixQ. / .s/ for real xand positive
which generate these algebras.
Moments of quantum Lévy areas 441
As mentioned above, the moments of Levy area are directly related to classical
formulas of Euler for values of the zeta function at even integers. Many more
modern applications depend on the so-called Levy area formula [12] for the
conditional characteristic function given the the values of the increments. Among
many recent results of this type are proofs of Apery’s theorem and more general
results on values of the zeta function at odd integers, and also new results on values
of multizeta functions. Because of mutual noncommutativity analogous joint
conditioning cannot be applied to the component processes of the quantum Levy
processes considered here. Instead, motivated by Yor’s conceptual simplification
of the conditional characteristic function using the rotational symmetry of planar
Brownian motion, which is shared in a certain sense by the quantum planar
Brownian motions considered here, it may be interesting to study what amounts
to a joint characteristic function for quantum Levy areas with its “radial part”
P.t/2CQ.t/2by regarding the latter as an iterated quantum stochastic integral.
Other quantum “quadratic Wiener functionals,” with their many mathematical
links, may also be explored. A start in this exploration has been made in [10].
The sticky shuffle Hopf algebra is reviewed in Section 2. Its use for reducing
the evaluation of moments to a counting problem is described in Section 3.
Several combinatorial results needed to accomplish the nontrivial counting task
are then provided in Section 4. These combinatorial results are crucial within
our calculations, but may become useful beyond the scope of this paper, too. In
Section 5, we evaluate the moments of the quantum Lévy area (10). Finally, in
Section 6, we show how the classical moments [11] are recovered in the “infinite
temperature” limit as ! 1:
2. The sticky shuffle product Hopf algebra
2.1. The shuffle product Hopf algebra. Given a complex vector space L, the
usual shuffle product Hopf algebra over Lis formed by equipping the vector space
T.L/DL1
nD0Nn
jD1Lof tensors of all ranks over Lwith the operations of
product, unit, coproduct, counit and antipode defined as follows. We denote a
general element ˛of T.L/by ˛D˛0˚˛1˚˛2˚ or .˛0; ˛1; ˛2; : : : /; where only
finitely many of the ˛mare nonzero. For each ˛m2Nm
jD1Lthe corresponding
embedded element .0; 0; : : : ; ˛m; 0; : : :/ of T.L/is denoted by ¹˛mº.
442 R. Hudson, U. Schauz, and Y. Wu
The shuffle product is defined by bilinear extension of the rule
¹L1˝L2˝ ˝ Lmº¹LmC1˝LmC2˝ ˝ LmCnº
DX
s2S.m;n/
¹Ls.1/ ˝Ls.2/ ˝ ˝ Ls.mCn/º(11)
where S.m; n/ denotes the set of .m; n/-shuffles, that is permutations sof
¹1; 2; : : : ; m Cnºfor which s.1/ < s.2/ < <s.m/ and s.m C1/ <
s.m C2/ < <s.m Cn/:
The unit element for this product is 1T.L/D.1C; 0; 0; : : : /.
The coproduct is the map from T.L/to T.L/˝T.L/defined by linear
extension of the rules that .1T.L//D1T.L/˝1T.L/D1T.L/˝T.L/and
¹L1˝L2˝ ˝ Lmº
D1T.L/˝ ¹L1˝L2˝ ˝ Lmº
C
m
X
jD2
¹L1˝L2˝ ˝ Lj1º ˝ ¹Lj˝LjC1˝ ˝ Lmº
C ¹L1˝L2˝ ˝ Lmº ˝ 1T.L/:
(12)
The counit "is the map from T.L/to Cdefined by linear extension of
".1T.L//D1C; " ¹L1˝L2˝ ˝ LmºD0for m > 0: (13)
The antipode is the map Sfrom T.L/to T.L/defined by linear extension of
S.1T.L//D1T.L/(14)
and
S¹L1˝L2˝ ˝ LmºD.1/m¹Lm˝Lm1˝ ˝ L1º(15)
for m > 0:
There are two useful equivalent definitions of the shuffle product. For the first,
we use the notational convention that, for arbitrary elements ˛of T.L/and Lof
L; ˛ ˝Lis the element of T.L/for which .˛ ˝L/0D0and .˛ ˝L/nD˛n1˝L
for n1: Then the shuffle product of arbitrary elements of T.L/is defined
inductively by bilinear extension of the rules
1T.L/¹L1˝L2˝ ˝ LmºD¹L1˝L2˝ ˝ Lmº1T.L/
D¹L1˝L2˝ ˝ Lmº;
(16)
Moments of quantum Lévy areas 443
and
¹L1˝L2˝ ˝ Lmº ¹LmC1˝LmC2˝ ˝ LmCnº
D.¹L1˝L2˝ ˝ Lm1º ¹LmC1˝LmC2˝ ˝ LmCnº/˝Lm
C.¹L1˝L2˝ ˝ Lmº ¹LmC1˝LmC2˝ ˝ LmCn1º/˝LmCn:
(17)
Here the two terms on the right-hand side of (17) correspond to the mutually
exclusive and exhaustive possibilities that s.m Cn/ Dmand s.m Cn/ DmCn
in the expansion (11). The second alternative definition is that the shuffle product
D˛ˇ of arbitrary elements ˛and ˇis given by
NDX
A[BD¹1;2;:::;N º
A\BD;
˛A
jAjˇB
jBj:(18)
Here the sum is over the 2Nordered pairs .A; B/ of disjoint subsets whose union is
¹1; 2; : : :; N ºand the notation is as follows; jAjdenotes the number of elements in
the set Aso that ˛jAjdenotes the homogeneous component of rank jAjof the tensor
˛D.˛0; ˛1; ˛2; : : : /; and ˛A
jAjindicates that this component is to be regarded as
occupying only those jAjcopies of Lwithin NN
jD1Llabelled by elements of the
subset Aof ¹1; 2; : : :; Nº:Thus with ˇB
jBjdefined analogously the combination
˛A
jAjˇB
jBjis a well-defined element of NN
jD1L.
2.2. The sticky shuffle algebra. Now suppose that the complex vector space L
is an associative algebra. We define the sticky shuffle product in the vector space
T.L/by modifying definition (17) by inserting an extra term so that now
¹L1˝L2˝ ˝ Lmº ¹LmC1˝LmC2˝ ˝ LmCnº
D.¹L1˝ ˝ Lm1º ¹LmC1˝ ˝ LmCnº/˝Lm
C.¹L1˝ ˝ Lmº ¹LmC1˝ ˝ LmCn1º/˝LmCn
C.¹L1˝ ˝ Lm1º ¹LmC1˝ ˝ LmCn1º/˝LmLmCn:
(19)
The sticky shuffle (also known as quasishuffle and as stuffle) Hopf algebra appears
to originate in [14]. Or we can modify the alternative definition of the shuffle
product (18) by defining the product D˛ˇ by
NDX
A[BD¹1;2;:::;N º
˛A
jAjˇB
jBj:(20)
444 R. Hudson, U. Schauz, and Y. Wu
Here the sum is now over the 3Nnot necessarily disjoint ordered pairs .A; B/
whose union is ¹1; 2; : : :; N º; ˛A
jAjand ˇB
jBjare defined as before but now if
A\B¤ ; double occupancy of a copy of Lwithin Nn
jD1Lis reduced to single
occupancy by using the multiplication in the algebra Las a map from LLto
L:Thus the sticky shuffle product reduces to the usual shuffle product in the case
when the multiplication in Lis trivial with all products vanishing. That (20) is
equivalent to (19) (and in particular, that (18) is equivalent to (17)) is seen by noting
that the three terms on the right-hand side of (19) correspond to the three mutually
exclusive and exhaustive possibilities that N2A\Bc; N 2Ac\Band N2A\B
in (20).
The same unit, coproduct and counit as before can be applied to make the sticky
shuffle product algebra into a Hopf algebra, but the definition of the antipode must
be modified [6] to
.1/mS¹L1˝L2˝ ˝ Lmº
D¹Lm˝Lm1˝ ˝ L1º
C
m
X
rD1X
1k1<k2<<kr1<m
¹Lkr1C1Lkr1C2: : : Lm
˝Lkr2C1Lkr2C2: : : Lkr1˝
˝L1L2: : : Lk1º:
(21)
The recovery formula [1] expresses the homogeneous components of an ele-
ment ˛of T.L/in terms of the iterated coproduct .N/˛by
˛ND..N/˛/
.1;1;:::;
.N /
1 /
:(22)
Here, .N/ is defined recursively by
.2/ Dand .N/ D. ˝Id˝.N 2/T.L//ı.N1/ for N > 2: (23)
Hence, it is a map from T.L/to the Nth tensor power
O.N/T.L/DO.N/
1
M
nD0
n
O
jD1
LD
1
M
n1;n2;:::;nND0
N
O
rD1
nr
O
jrD1
L(24)
so that .N /˛has multirank components ˛.n1;n2;:::;nN/of all orders. The recovery
formula (21) also holds when ND0and ND1if we define .0/ and .1/ to be
the counit "and the identity map IdT.L/respectively.
Moments of quantum Lévy areas 445
Note that the recovery formula is the same for both the sticky and nonsticky
cases; it only involves the coproduct which is one and the same map. However
our application of it will use the fact that is multiplicative, .˛ˇ/ D.˛/.ˇ/,
where the product on the tensor square T.L/˝T.L/is defined by linear extension
of the rule
.a ˝a0/.b ˝b0/Dab ˝a0b0:(25)
This holds in particular with the sticky shuffle product as the product in T.L/.
3. Moments and sticky shuffles
We now describe the connection between sticky shuffle products and iterated
stochastic integrals. We begin with the well-known fact that, for the one-dimen-
sional Brownian motion Xand for ab;
.X.b/ X.a//2D2Z
ax<b
.X.x/ X.a//dX.x/ CZ
ax<b
dT .x/; (26)
where T .x/ Dxis time. We introduce the Itô algebra LDChdX; dT iof complex
linear combinations of the basic differentials dX and dT; which are multiplied
according to the table
dX dT
dX
dT
dT 0
0 0
(27)
together with the corresponding sticky shuffle Hopf algebra T.L/: For each pair of
real numbers a < b; we introduce a map Jb
afrom T.L/to complex-valued random
variables on the probability space of the Brownian motion Xby linear extension
of the rule that, for arbitrary dL1; dL2;dLm2¹dX; dT º
Jb
a¹dL1˝dL2˝ ˝ dLmº
DZ
ax1<x2<<xm<b
dL1.x1/dL2.x2/dL3.x3/dLm.xm/
D
b
Z
a
x4
Z
a
x3
Z
a
x2
Z
a
dL1.x1/dL2.x2/dL3.x3/dLm.xm/:
(28)
By convention Jb
amaps the unit element of the algebra T.L/to the unit random
variable identically equal to 1.
446 R. Hudson, U. Schauz, and Y. Wu
Then (26) can be restated as follows,
Jb
a.¹dXº/J b
a.¹dXº/DJb
a.¹dXº ¹dXº/; (29)
using the fact that ¹dXº2D2¹dX ˝dXºC¹dT º:
The following more general Theorem is probably known to many classical and
quantum probabilists; the quantum version was first given in [2].
Theorem 4. For arbitrary ˛and ˇin T.L/,
Jb
a.˛/J b
a.ˇ/ DJb
a.˛ˇ/:
Proof. By bilinearity it is sufficient to consider the case when
˛D ¹dL1˝dL2˝ ˝ dLmº; ˇ D ¹dLmC1˝dLmC2˝ ˝ dLmCnº(30)
for dL1; dL2; ; dLmCn2¹dX; dT º:In this case Theorem 4follows, using the
inductive definition (19) for the sticky shuffle product, from the product form of
Itô’s formula,
d./ D.d/ Cd C.d/d (31)
where stochastic differentials of the form d DFdX CGdT; with stochastically
integrable processes Fand G, are multiplied using table (27).
For planar Brownian motion RD.X; Y / the Ito table (27) becomes
dX dY dT
dX dT 0 0
dY 0 dT 0
dT 000
(32)
For the quantum planar Brownian motion .P ./; Q.//it becomes
dP ./ dQ./ dT
dP ./ 2dT idT 0
dQ./ idT 2dT 0
dT 0 0 0
(33)
Theorem 5. Theorem 4holds when Lis either of the algebras defined by the
multiplication tables (32) and (33).
Remark 6. In both cases this follows from the corresponding Itô product rule (31).
For classical planar Brownian motion the Itô product rule is well-known. For the
quantum case, when D1see [15] and when > 1, see [8].
Moments of quantum Lévy areas 447
In view of (8) and (9)
AŒa;b/ D1
2Jb
a.dX ˝dY dY ˝dX/; (34)
B./
Œa;b/ D1
2Jb
a.dP ./ ˝dQ./ dQ./ ˝dP .//(35)
For use below we note that the table (32) becomes
dZ d x
Z dT
dZ 01
2dT 0
dx
Z1
2dT 0 0
dT 0 0 0
(36)
in terms of the basis .dZ; d x
Z; dT / where
dZ D1
2.idX CdY /; d x
ZD1
2.idX CdY /: (37)
Correspondingly
AŒa;b/ DiJ b
a.dZ ˝dx
Zdx
Z˝dZ/ (38)
Similarly (33) becomes
dA./ dA./ dT
dA./ 01
2.2C1/dT 0
dA./ 1
2.21/dT 0 0
dT 0 0 0
(39)
in terms of the basis .dA./; dA./; dT / where
dA./ D1
2.idP ./ CdQ.//; dA./ D1
2.idP ./ CdQ.//: (40)
For the basis .d y
P./; d y
Q./; dT /; (33) becomes
dy
P./ dy
Q./ dT
dy
P./ dT i2dT 0
dy
Q./ i2dT dT 0
dT 0 0 0
(41)
which deforms to the classical table (32) as ! 1:Similarly, for the basis
.d y
A./; d y
A./; dT / where
dy
A./ D1
2.id y
P./ Cdy
Q.//; d y
A./ D1
2.id y
P./ Cdy
Q.//(42)
448 R. Hudson, U. Schauz, and Y. Wu
we have
dy
A./ dy
A./ dT
dy
A./ 0 CdT 0
dy
A./ dT 0 0
dT 0 0 0
(43)
with
˙D1
2.1 ˙2/; (44)
which becomes isomorphic to (36) when ! 1:The normalized quantum Lévy
area which is our main concern is
y
B./
Œa;b/ DiJ b
a.d y
A./ ˝dy
A./ dy
A./ ˝dy
A.//: (45)
In the following theorem, the basis referred to is any of those for which the
respective algebras have multiplication tables (32), (33), (36), (39), (41) or (43).
Theorem 7. For arbitrary n2N; a < b 2Rand basis elements dL1,dL2, ...,
dLn,
EŒJ b
a¹dL1˝dL2˝ ˝ dLnºD0
unless
dL1DdL2D D dLnDdT:
Proof. If dLn¤dT then
Jb
a¹dL1˝dL2˝ ˝ dLnºD
b
Z
a
Jx
a¹dL1˝dL2˝ ˝ dLn1ºdLn.x/: (46)
In the classical cases (32) and (36), Lnis a real or complex-valued martingale and
the expectation of the stochastic integral against dLnvanishes. When D1it
also vanishes in the cases (33), (39) and (41) by the first fundamental formula of
quantum stochastic calculus in the Fock space F[15]. When > 1 it vanishes as
may be seen for example by realising the processes P./ and Q./ in the tensor
product of Fwith its Hilbert space dual, F˝x
F, equipped with the tensor product
e.0/ ˝e.0/ of the Fock vacuum vector with its dual vector as
P./ Dr1
2.2C1/.P ./ ˝x
I/ Cr1
2.21/.I ˝x
P.//; (47)
Q./ Dr1
2.2C1/.Q./ ˝x
I/ Cr1
2.21/.I ˝x
Q.//; (48)
Moments of quantum Lévy areas 449
and again invoking the first fundamental formula. Thus in all cases
EŒJ b
a¹dL1˝dL2˝ ˝ dLnºD0(49)
unless dLnDdT:
If dLnDdT then by Fubini’s theorem we can write
EŒJ b
a¹dL1˝dL2˝ ˝ dLnº
D
b
Z
a
¹EŒJ x
a¹dL1˝dL2˝ ˝ dLn1ººdx
D0;
(50)
unless dLn1DdT , by the previous argument. By repetition we see that
EŒJ b
a¹dL1˝dL2˝ ˝ dLnºD0; (51)
unless each of dLn; dLn1,dLn2,..., dL1is equal to dT .
Now consider the moments sequence of the normalized quantum Lévy area of
variance 2in the form (45). In view of Theorem 4,
Œy
B./
Œa;b/nDin.J b
a.d y
A./ ˝dy
A./ dy
A./ ˝dy
A.///n
DinJb
a.¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./ºn/
(52)
The nth sticky shuffle power ¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./ºnwill consist
of non-sticky shuffle products of rank 2n together with terms of lower ranks
n; n C1; : : : ; 2n 1, all of which except the rank nterm will contain one or more
copies of dy
A./ and dy
A./;and will thus not contribute to the expectation in
view of Theorem 7. The term of rank nwill be a multiple of dT ˝dT ˝
.n/
dT ,
where the symbol .n/ on the top of the last copy dT indicates the position of the
referred term in a row. Thus we can write
¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./ºn
Dw./
n¹dT ˝dT ˝
.n/
dT º C terms of rank > n:
(53)
450 R. Hudson, U. Schauz, and Y. Wu
for some coefficient w./
n:The corresponding moment is given by
EŒy
B./
Œa;b/nDinw./
nEŒJ b
a.¹dT ˝dT ˝
.n/
dT º/
Dinw./
nZ
ax1<x2<<xn<b
dx1dx2: : : dxn
Dinw./
n
.b a/n
nŠ :
(54)
By the recovery formula (21) and the multiplicativity of the nth order coprod-
uct .n/;
w./
ndT ˝dT ˝
.n/
dT
D ¹¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./ºnºn
D..n/.¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./ºn//
.1;1;:::;
.n/
1 /
D...n/.¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./º//n/
.1;1;:::;
.n/
1 /
:
(55)
Now
.n/.¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./º/
DX
1jn
1T.L/˝ ˝
.j /
¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./º ˝ ˝ 1T.L/
CX
1j <kn
.1T.L/˝ ˝
.j /
¹dy
A./º ˝ ˝
.k/
¹dy
A./º ˝ ˝
.n/
1T.L/
1T.L/˝ ˝
.j /
¹dy
A./º ˝ ˝
.k/
¹dy
A./º ˝ ˝
.n/
1T.L//:
(56)
The first term of this sum, being of rank 2; cannot contribute to the component of
joint rank .1; 1; : : : ;
.n/
1 / of the nth power of .n/.¹dX ˝dY dY ˝dXº/, where
product in the nth tensor power N.N/T.L/is defined exactly as in the case nD2
Moments of quantum Lévy areas 451
in (25). Thus
w./
ndT ˝dT ˝
.n/
dT
D...n/.¹dy
A./ ˝dy
A./ dy
A./ ˝dy
A./º//n/
.1;1;:::;
.n/
1 /
D X
1j <kn
.1T.L/˝ ˝
.j /
¹dy
A./º ˝ ˝
.k/
¹dy
A./º ˝ ˝
.n/
1T.L/
1T.L/˝ ˝
.j /
¹dy
A./º ˝ ˝
.k/
¹dy
A./º ˝ ˝
.n/
1T.L//n.1;1;:::;
.n/
1 /
(57)
This calculation of w./
ncan be finished using some combinatorics. We do that in
the following two sections.
4. Some background about Eulerian and Euler numbers
In this section, we present several lemmas about Euler numbers, Eulerian num-
bers, Euler polynomials and forth-back permutations, as we call them. These com-
binatorial results are of sufficient general nature to be of interest elsewhere. A view
of the provided lemmas appear as exercises in [16], but are still proven here for the
sake of completeness. Additional explanations to the used methods and many
similar results can be found in [16,17].
A permutation sin the symmetric group Snis a zigzag permutation (mislead-
ingly also called alternating permutation) if s.1/ > s.2/ < s.3/ > s.4/ < .
In other words, sis zigzag if s.1/ > s.2/ and
either s.j 1/ < s.j / > s.j C1/ or s.j 1/ > s.j / < s.j C1/ (58)
for all j2 ¹2; 3 : : :; n 1º. If we have the initial condition s.1/ < s.2/, instead of
s.1/ > s.2/, we may call szagzig. The number of all zigzag permutations in Snis
the Euler zigzag number An. These numbers occur in many places, for instance,
as the coefficients of z2n
.2n/Š in the Maclaurin series of sec.z/Ctan.z/. In this paper,
we meet them as the number of forth-back permutations, as we call them. These
are the permutations s2Snwith
either s1.j / < j > s.j / or s1.j / > j < s.j / (59)
for all j2 ¹1; 2; : : :; nº. As forth-back permutations do not contain cycles of odd
length, nmust be even for there to exist forth-back permutations. If nis even, say
nD2m > 0, we have the following lemma:
452 R. Hudson, U. Schauz, and Y. Wu
Lemma 8. The number of forth-back permutations in S2m is the Euler zigzag
number A2m.
Proof. A bijection between the forth-back permutations sand the zigzag permu-
tations in S2m is obtained by applying the so-called transformation fundamen-
tale [4]. To perform this transformation, we write sin cycle notation
sD.s1;s2; : : : ; s`21/.s`2;s`2C1; : : : ; s`31/.s`3;s`3C1; : : : ; s`41/
.s`m;s`mC1; : : : ; s2m/: (60)
This representation and the numbers sjare uniquely determined if we require that
the first entry of every cycle is bigger than all other entries in that cycle, and also
that s1<s`2<s`3< <s`m. The new permutation N
sis then obtained by
forgetting brackets and setting N
s.j / WD sj. We just have to see that this actually
yields a bijection s7! N
sbetween forth-back and zigzag permutations. To do this
we proceed as follows.
Assume first that sis forth-back. Then all cycles necessarily have even length
and the permutation N
sis obviously zigzag, s1>s2<s3>s4< >s2m.
Conversely, let us show that every zigzag permutation N
shas a unique pre-image
s, and that that pre-image is forth-back. To construct a pre-image sof N
s, we only
need to find suitable numbers `j, which indicate where we have to insert brackets
into the sequence .s1;s2; : : : ; s2m/WD .s.1/; s.2/; : : :; s.2m// to actually get a pre-
image. However, if we have already found `2; `3; : : : ; `j, then `jC1is necessarily
the first index xwith sx>s`j. Using this, we can construct a pre-image sof N
sin
S2m, and it is uniquely determined. Moreover, if N
sis zigzag then this construction
ensures that s1and the s`jare peaks and their neighbors and s2m are valleys.
Since also s1>s`21,s`2>s`31, ..., s`m>s2m, insertion of brackets before
the peaks `jyields forth-back cycles in s.
With the bijection established, it is now clear that there are as many forth-back
permutations as there are zigzag permutations in S2m. This number is the Euler
zigzag number A2m.
The number of forth-back permutations with just one cycle is given by the
following lemma. If Cndenotes the subset of cyclic permutations in Sn, we have:
Lemma 9. The number of forth-back permutations in C2m is A2m1.
Proof. The cycle notation sD.s1;s2; : : :; s2m/of cyclic permutations s2S2m
is not uniquely determined, as one may rotate the entries cyclically. It becomes
uniquely determined if we require that s2m D2m. In this case, removal of the
Moments of quantum Lévy areas 453
last entry yields a sequence .s1;s2; : : : ; s2m1/that is zagzig (with s1<s2as
s2m was the biggest entry of s). If we define N
s2S2m1by setting N
s.j / WD sj,
for jD1; 2; : : :; 2m 1, we obtain a bijection s7! N
sfrom the cyclic forth-back
permutations in S2m to the zagzig permutations in S2m1. Indeed, every zagzig
permutation N
sin S2m1has the cycle sWD .N
s.1/; N
s.2/; : : :; N
s.2m1/; 2m/ as unique
pre-image. The existence of this bijection shows that the number of cyclic forth-
back permutations in S2m is equal to the number of zagzig permutations in S2m1,
which is A2m1, as for zigzag permutations.
This enumerative result about cyclic forth-back permutations can also be ap-
plied to forth-back permutation with kcycles of lengths 2m1; 2m2; : : : ; 2mk(nec-
essarily all even). To formulate this, we denote with Cn1;n2;:::;nkthe set of all per-
mutations in Snwith kcycles of lengths n1; n2; : : : ; nk, i.e. the set of permutations
of typ .n1; n2; : : : ; nk/, as we say. We also denote with n
n1;n2;:::;nkthe number of
unordered partitions ¹N1; N2; : : : ; Nkºof the set ¹1; 2; : : :; nºwith kblocks Njof
sizes jNjj D nj> 0. With this we get the following more general formula.
Lemma 10. If positive integers m1m2 mkwith m1Cm2C:::CmkDm
are given, then the number of forth-back permutations in S2m with kcycles of
lengths 2m1; 2m2; : : : ; 2mkis
j¹s2C2m1;2m2;:::;2mkjsis forth-backºj D 2m
2m1; 2m2; : : : ; 2mkk
Y
jD1
A2mj1:
In particular,
A2m DX2m
2m1; 2m2; : : : ; 2mkk
Y
jD1
A2mj1;
where the sum runs over all partitions m1Cm2C C mkof m, that is over
all non-decreasing sequences m1m2 mkof positive integers of every
possible length kwith m1Cm2C C mkDm > 0.
Proof. If a partition m1Cm2C C mkDmof the number mis given, then
there are 2m
2m1;2m2;:::;2mkpartitions of the set ¹1; 2; : : :; 2mºinto a set of kblocks
Njwith sizes 2mj,jD1; 2; : : : ; k. The block Njcan be turned into a cyclic
forth-back permutation in A2mj1many ways, by Lemma 9. Hence, we get the
stated expression for the number of forth-back permutations of that type.
454 R. Hudson, U. Schauz, and Y. Wu
Moreover, it is easy to see that the sum over all possible expressions of this
form gives the number of all forth-back permutations, which is A2m by Lemma 8.
Indeed, every forth-back permutations sin S2m, has a certain number kof cycles
and a certain type, certain lengths 2m1; 2m2; : : :; 2mkof its cycles. In this respect,
every partition m1Cm2C C mkDmis possible. Hence, the sum covers all
A2m forth-back permutations, as stated.
In our investigations, we will also need to look at a certain notion of sign,
denoted by sn.s/, for permutations s2Sn, defined by
sn.s/WD
n
Y
jD1
.1/des.j;s.j //;(61)
where
des.h; k/ WD ´0if hk;
1if h > k: (62)
We want to show that Ps2S¤
2m
sn.s/D.1/mA2m, where S¤
ndenotes the set of
fixed-point-free permutations in Sn. To establish this and similar results, we need
to introduce certain equivalence classes of permutations which are based on the
notion of a transit of a permutation. We call h2¹1; 2; : : : ; nºatransit of the
permutation s2Snif
either s1.h/ < h < s.h/ or s1.h/ > h > s.h/ : (63)
Let ST
ndenote the set of permutations in Snwhich contain a transit, and let
CT
n1;n2;:::;nkbe the set of permutations in Cn1;n2;:::;nkwhich contain a transit. Every
permutation swith transit contains a unique smallest transit h, say inside a cycle
.j1; j2; : : : ; s1.h/; h; s.h/; : : :; j`/of length `, which we may also write as
j17! j27! 7! s1.h/ 7! h7! s.h/ 7! 7! j`7! j1:(64)
We obtain a permutation s0of ¹1; 2; : : :; nº n ¹hºby replacing the chain of assign-
ments s1.h/ 7! h7! s.h/ with the shorter chain s1.h/ 7! s.h/. Hence, the
new permutation s0contains the cycle
j17! j27! 7! s1.h/ 7! s.h/ 7! 7! j`7! j1:(65)
We define an equivalence relation on the set ST
n. For two permutations sand r
with transit, we write srif and only if sand rhave the same smallest transit
h, if s0Dr0and if the smallest transit his missing from the same cycle in s0as
in r0. The equivalence class of sis denoted as Œs. The equivalence relation can
also be restricted to the sets of the form CT
n1;n2;:::;nk. We have ŒsCT
n1;n2;:::;nk
whenever s2CT
n1;n2;:::;nk.
Moments of quantum Lévy areas 455
Example 11. In the 8-cycle sWD .4; 1; 8; 2; 6; 7; 5; 3/ the number 5is the smallest
transit, a downwards transit in this case, as 7 > 5 > 3. If we remove it from
the cycle, and reinsert the 5as a transit in all possible ways into the remaining
7-cycle .4; 1; 8; 2; 6; 7; 3/, we get four permutations. The 5would not be a transit
between 4and 1, but can be inserted between 1and 8, yielding .4; 1; 5; 8; 2; 6; 7; 3/.
Similarly, we also obtain .4; 1; 8; 5; 2; 6; 7; 3/,.4; 1; 8; 2; 5; 6; 7; 3/ and the original
permutation .4; 1; 8; 2; 6; 7; 5; 3/. These four 8-cycles form the equivalence class
Œsof swith respect to . Interestingly, two of the four 8-cycles contain the number
5as upwards transit, and their sign is opposite to the sign of the other two 8-cycles
with 5as downwards transit, as one can easily check. The situation is illustrated
in Figure 1.
Figure 1. The cycle .4; 1; 8; 2; 6; 7; 5; 3/ with smallest transit hD5.
The observation about the sign of the elements in equivalence classes that we
made in this example is no coincidence. We have the following lemma.
Lemma 12. If a permutation scontains a transit, then there is an even number
of elements in its equivalence class Œs. One half of them have negative sign, and
one half have positive sign.
Proof. Assume the cycle that contains the smallest transit hof sis denoted as in
equation (64). If we walk once around the shortened cycle of s0in equation (65)
and observe the indices j1; j2; j3; : : : as a kind of altitude, then we will cross the
altitude has many times upwards (from below hto above h) as downwards (from
above hto below h), see Figure 1. Hence, there are as many ways to reinsert
456 R. Hudson, U. Schauz, and Y. Wu
has an upwards transit and as a downwards transit. Therefore, one half of the
permutations that we obtain have positive sign, and one half have negative sign.
The equivalence class Œsof sis as claimed.
With the help of Lemma 12, we can now prove the following theorem.
Theorem 13. Let a partition n1Cn2C:::CnkDnwith 2n1n2 nk
be given. Then,
X
s2Cn1;n2;:::;nk
sn.s/D´.1/n
2n
n1;n2;:::;nkQk
jD1Anj1if all njare even,
0otherwise.
In particular,
X
s2S¤
n
sn.s/D´.1/n
2Anif nis even,
0otherwise.
Proof. We observe that we can cancel out all permutations in Cn1;n2;:::;nkthat
contain a transit, that is, all elements of CT
n1;n2;:::;nk. In fact, CT
n1;n2;:::;nkis parti-
tioned into equivalence classes, and each of them cancels out by Lemma 12. The
remaining elements of Cn1;n2;:::;nkdo not contain a transit. Hence, if there are
any remaining permutations in Cn1;n2;:::;nknCT
n1;n2;:::;nk, they must be forth-back
permutations. In particular, in this case, all njmust necessarily be even. Now,
Lemma 10 yields the first stated result, since all forth-back permutations in S¤
n
have sign .1/n
2. The second formula follows from this and the second formula
in Lemma 10, but it can also be deduced from Lemma 8directly, after canceling
out equivalence classes in S¤
n.
For the number of partitions in some of the previous results, we also have the
following well known formula, whose proof we present for completeness:
Lemma 14. We have
n
n1; n2; : : : ; nkDn
n1; n2; : : :; nk1
k1Šk2ŠkrŠDnŠ
n1Šn2ŠnkŠk1Šk2Š : : : krŠ;
where k1; k2; : : : ; krare the multiplicities of the different elements in the multi-set
¹n1; n2; : : :; nkº. (For example, the elements in the multi-set ¹2; 2; 2; 4; 4ºhave
the multiplicities k1D3and k2D2, and rD2.)
Moments of quantum Lévy areas 457
Proof. Without loss of generality, we may assume that n1n2 nk.
There are n
n1;n2;:::;nkWD nŠ
n1Šn2ŠnkŠordered partitions (sequences of blocks) with
block sizes n1; n2; : : : ; nk(in that order). This number can also be generated
by first choosing all n
n1;n2;:::;nkunordered partitions (sets of blocks) with block
sizes n1; n2; : : : ; nk, and then arranging each of them in all possible ways as a
sequences of blocks, i.e. as ordered partition. Hence, for each unordered partition
¹N1; N2; : : :; Nkº, we have to see how many ways there are to arrange its blocks
in a sequence with nondecreasing cardinalities (equal to the sequence n1
n2 nk). Ambiguities in this order of the blocks are only given for
blocks of equal size, which correspond to multiplicities of the elements in the
multi-set ¹n1; n2; : : : ; nkº. Hence, the number of ways is always k1Šk2Š : : : krŠ,
where k1; k2; : : : ; krare the multiplicities of the different elements in the multi-set
¹n1; n2; : : :; nkº. Combining this factor with the number of unordered partitions
yields the relation
n
n1; n2; : : : ; nkk1Šk2ŠkrŠDn
n1; n2; : : : ; nk;(66)
which proves the lemma.
In this paper we will also consider the number of descends of sequences
.j1; j2; : : : ; jn/of n1integers, that is, the number
des.j1; j2; : : : ; jn/WD j¹`2 ¹1; 2; : : : ; n 1ºjj`> j`C1ºj ;(67)
which generalizes des.h; k/ 2 ¹0; 1ºin (62). The number of permutations s2Sn
for which
des.s.1/; s.2/; : : : ; s.n// Dj(68)
is the so-called Eulerian number ˝n
j˛. We follow [16] in taking this as the definition
of the Eulerian numbers, but Eulerian numbers also count various other kinds of
objects, see [16]. The corresponding generating function is the so-called Euler
polynomial
Sn./ WD X
s2Sn
des.s.1/;s.2/;:::;s.n// D
n1
X
jD0n
jj:(69)
In this paper, we will also need the closely related number of cyclic descends,
defined by
cdes.j1; j2; : : : ; jn/WD des.j1; j2; : : : ; jn; j1/; (70)
which has the following statistic [16, Exercise1.11].
458 R. Hudson, U. Schauz, and Y. Wu
Lemma 15. All permutations s2Snhave 0 < cdes.s.1/; s.2/; : : :; s.n// < n. For
0 < j < n, the number of permutations s2Snwith exactly jcyclic descents is
j¹s2Snjcdes.s.1/; s.2/; : : :; s.n// Djºj D nn1
j1:
In particular,
X
s2Sn
cdes.s.1/;s.2/;:::;s.n// DnSn1./: (71)
Proof. Assume s2Snwith cdes.s.1/; s.2/; : : :; s.n// Dj. Since every sequence
of distinct integers has at least one cyclic descent and one cyclic ascent, jcannot
be 0or n, and 0 < j < n as claimed. Now, let Mbe such that s.M/ is the biggest
entry of the sequence .s.1/; s.2/; : : :; s.n//. We construct a new shorter sequence
N
sWD .s.MC1/; s.M C2/; : : : ; s.n/; s.1/; s.2/; : : :; s.M 1// (72)
by removing s.M/ and gluing together the remaining halves in opposite order.
Obviously, N
shas exactly j1descends. If we first rotate the entries of the
sequence .s.1/; s.2/; : : : ; s.n// and then remove the biggest entry, we still get the
same sequence N
sin the same way. This idea shows that removal of the biggest
entry yields an nto 1correspondence s7! N
sbetween the permutations in Snwith
jcyclic descends and the permutations in Sn1with j1descends. Hence,
there are n˝n1
j1˛permutations in Snwith exactly jcyclic descends. In particular,
this number is the coefficient of jin both polynomial on the left of (71) and the
polynomial on the right. So these polynomials are equal.
One can also prove the following lemma, which might be useful in calculations
similar to the ones in our paper [16, Excercise 1.7].
Lemma 16. The number of permutations s2Snwith s.x/ < x for exactly jpoints
x2 ¹1; 2; : : :; nºis the Eulerian number ˝n
j˛and the corresponding generating
function is the Euler polynomial.
5. Moments of quantum Lévy areas
To evaluate the moments EŒy
B./
Œa;b/n, we need to calculate the number w./
n, as
explained in (54). By (57), we have
w./
ndT ˝dT ˝
.n/
dT DX
h¤k
.1/des.h;k/Rh;kn.1;1;:::;
.n/
1 / (73)
Moments of quantum Lévy areas 459
with
Rh;k WD 1˝ ˝ 1˝
.h/
¹dy
A./º ˝ 1˝ ˝ 1˝
.k/
¹dy
A./º ˝ 1˝ ˝ 1: (74)
As in the previous section,
des.h; k/ WD ´0if hk;
1if h > k; (75)
and the nth power is based on the sticky shuffle product in T.L/and its extension
to the nth tensor power N.N/T.L/, as described in (25)for nD2.
If we set eWD .h; k/, then we may also write Refor Rh;k and des.e/ for
des.h; k/. Using distributivity, this yields
w./
ndT ˝dT ˝
.n/
dT DX
n
Y
`D1
.1/des.e`/Re`.1;1;:::;
.n/
1 /
;(76)
where the sum runs over all n-tuples .e1; e2; : : : ; en/of pairs .h; k/ with h¤k.
We may imagine each pair e`D.h`; k`/as a directed edge, an arc, from h`to k`.
Each n-tuples .e1; e2; : : : ; en/is then a directed labeled multigraph, or digraph, on
the vertex set VWD ¹1; 2; : : :; nº. It is important to keep track of the indices `as
labels of the arcs e`, because our product is not commutative,
dy
A./dy
A./ DCdT and dy
A./dy
A./ DdT ; (77)
and
.¹dy
A./º¹dy
A./º/.1/ DCdT and .¹dy
A./º¹dy
A./º/.1/ DdT : (78)
For example, in the case nD4, the two arcs e1D.1; 2/ and e2D.2; 3/ contribute
.Re1Re2/.1;1;1;1/
D..¹dy
A./º ˝ ¹dy
A./º ˝ 1˝1/.1 ˝ ¹dy
A./º ˝ ¹dy
A./º ˝ 1//.1;1;1;1/
D.¹dy
A./º1/.1/ ˝.¹dy
A./º¹dy
A./º/.1/ ˝.1¹dy
A./º/.1/ ˝.1 1/.1/
Ddy
A./ ˝dT ˝dy
A./ ˝1;
(79)
while if the labels 1and 2are exchanged, e1D.2; 3/ and e2D.1; 2/, we get
.Re1Re2/.1;1;1;1/
D..1 ˝ ¹dy
A./º ˝ ¹dy
A./º ˝ 1/.¹dy
A./º ˝ ¹dy
A./º ˝ 1˝1//.1;1;1;1/
DCdy
A./ ˝dT ˝dy
A./ ˝1:
(80)
460 R. Hudson, U. Schauz, and Y. Wu
In order to calculate the coefficient w./
nof dT ˝dT ˝ ˝ dT in (73),
we need to retain only those summands in (76)that contribute scalar multiple of
dT ˝dT ˝ ˝ dT . We may discard other summands. Hence, we do not have
to sum over all digraphs .e1; e2; : : : ; en/. To see which ones we have to retain,
let us assume that .e1; e2; : : : ; en/yields a multiple of dT ˝dT ˝ ˝ dT
in (76). Since the ncopies of dy
A./ and ncopies of dy
A./ in the unexpanded
product Qn
`D1Re`must yield ncopies of dT , one in each possible position,
each vertex of the digraph .e1; e2; : : : ; en/must have exactly one incoming arc
and one outgoing arc. Thus, .e1; e2; : : : ; en/must consist of disjoint cyclically
oriented cycles that cover V. This allows us to view each arc e`D.h`; k`/as the
assignment of a function value, h`7! k`DW s.h`/. We obtain a fixed-point-free
permutation son VD ¹1; 2; : : :; nº. We obtain a second permutation lon Vby
assigning to each label `2Vthe vertex h`from which the arc e`D.h`; k`/
originates, `7! h`DW l.`/. The pair .l;s/of permutations, lin Snand sin
the set S¤
nof fixed-point-free permutations of VD ¹1; 2; : : :; nº, contains the
full information about .e1; e2; : : :; en/. Our construction describes a bijection
.e1; e2; : : : ; en/7! .l;s/from the set of digraphs .e1; e2; : : : ; en/that contribute a
multiple of dT ˝dT ˝ ˝dT onto the set SnS¤
n. The edges e`of the digraph
.e1; e2; : : : ; en/can be recovered from sand lthrough the formula
e`D.l.`/; s.l.`///; (81)
which describes the inverse bijection .l;s/7! .e1; e2; : : : ; en/. With this, the term
w./
ndT ˝dT ˝ ˝ dT in (76)can be calculated as
w./
ndT ˝dT ˝ ˝ dT
DX
.l;s/2SnS¤
n
n
Y
`D1
.1/des.l.`/;s.l.`///Rl.`/;s.l.`//.1;1;:::;
.n/
1 /
DX
.l;s/2SnS¤
n
sn.s/n
Y
`D1
Rl.`/;s.l.`//.1;1;:::;
.n/
1 /
;
(82)
where, for every l2Sn,
sn.s/WD
n
Y
jD1
.1/des.j;s.j // D
n
Y
`D1
.1/des.l.`/;s.l.`//:(83)
We have to consider the product Qn
`D1Rl.`/;s.l.`//, for every fixed .l;s/2
SnS¤
n. In this product, one dT is produced in each position j, and it comes either
with the scalar factor Cor with . If dT arrives as dy
A./dy
A./ then we get C
Moments of quantum Lévy areas 461
as scalar factor, if it arrives as dy
A./dy
A./ then we get . To examine how the
dT in position jarrives, let `1WD l1.j / and `2Dl1.s1.j //, then Rl.`1/;s.l.`1//
contributes a dy
A./ in position jas `1th factor, and Rl.`2/;s.l.`2// contributes a
dy
A./ in position jas `2th factor. So, if l1.s1.j /// > l1.j / then `2> `1and
the dy
A./ comes after the dy
A./, yielding a Cas scalar factor. In general, the
scalar factor of the dT in position jis .C=/des.l1.s1.j //;l1.j //. Therefore,
w./
nDn
X
.l;s/2SnS¤
n
sn.s/
n
Y
jD1
.C=/des.l1.s1.j //;l1.j //:(84)
We substitute s.j / for jand l1for l, and obtain the following theorem.
Theorem 17. We have
w./
nDn
X
s2S¤
n
sn.s/X
l2Sn
n
Y
jD1
des.l.j /;l.s.j ///;
where WD C=.
In this expression for w./
nthere are many terms that cancel against each other
when we carry out the sum. To remove these unnecessary summands and bundle
together equal terms, we first study the inner sum
ws
n./ WD X
l2Sn
n
Y
jD1
des.l.j /;l.s.j ///;(85)
for a fixed s2S¤
n. Initially, for simplicity, also assume that there is only one
cycle, of length nin s. In cycle notation, sD.s1;s2; : : :; sn/with s2Ds.s1/,
s3Ds.s2/, etc. In this particular case, by Lemma 15,
ws
n./ DX
l2Sn
n
Y
`D1
des.l.s`/;l.s`C1//
DX
l2Sn
cdes.l.s1/;l.s2/;:::;l.sn//
DX
r2Sn
cdes.r.1/;r.2/;:::;r.n//
DnSn1./;
(86)
where Sn./ is the Euler polynomial and cdes.r.1/; r.2/; : : :; r.n// denotes the
number of descends of the sequence .r.1/; r.2/; : : :; r.n/; r.1//.
462 R. Hudson, U. Schauz, and Y. Wu
Formula (86)holds only for cyclic permutations s2S¤
n. For the general
case, suppose that shas kDk.s/cycles of lengths n1; n2; : : : ; nk, say where
2n1n2 nkand n1Cn2C C nkDn. We say that .n1; n2; : : : ; nk/
is the typ of sand write s2Cn1;n2;:::;nk. We have to split the product in the
definition (85)of ws
n./ into kparts correspondingly. If C`denotes the set of the
n`elements of the `th cycle of the fixed given s2Cn1;n2;:::;nk, then
n
Y
jD1
des.l.j /;l.s.j /// D
k
Y
`D1Y
j2C`
des.l`.j /;l`.s.j ///;(87)
where l`is the restriction of lto C`, so that lDl1[l2[ [ ln. The range
of each l`can be any subset N`Vof n`elements, provided only that all the
subsets N`together form an ordered partition .N1; N2; : : : ; Nk/of V. We want to
describe the set Snof permutations lin terms of smaller bijections l`WC`!N`.
Let Ndenotes the set of all partitions NWD .N1; N2; : : : ; Nk/of Vinto kblocks
N`with jN`j D n`, let B`.N / be the set of bijections from C`to N`, and let
B.N / WD B1.N / B2.N / Bn.N /. With this, the set of permutations Snis
partitioned as
SnD[
N2N
¹l1[l2[ [ lnj.l1;l2; : : : ; ln/2B.N /º:(88)
From that disjoint union we get
ws
n./ DX
N2NX
l2B.N /
k
Y
`D1Y
j2C`
des.l`.j /;l`.s.j ///
DX
N2N
k
Y
`D1X
l`2B`.N/ Y
j2C`
des.l`.j /;l`.s.j ///:
(89)
Here, for all N2N, the inner sum is
X
l`2B`.N/ Y
j2C`
des.l`.j /;l`.s.j /// Dn`Sn`1./; (90)
by (86), because the names of the elements in C`and N`do not matter. Every
fixed set N`of n`different numbers is linearly ordered and produces the same
statistic for the cyclic descents, if we consider all sequences that can be arranged
using all elements of N`. Using
jNj D n
n1; n2; : : : ; nkWD nŠ
n1Šn2ŠnkŠ;(91)
Moments of quantum Lévy areas 463
we obtain
ws
n./ WD n
n1; n2; : : : ; nkk
Y
jD1
njSnj1./; (92)
where .n1; n2; : : : ; nk/is still the typ of s, i.e. s2Cn1;n2;:::;nk.
We can now calculate w./
nout of (92)and Theorem 17. Since we have the
disjoint union
S¤
nD[
n1Cn2CCnkDn;
2n1n2nk
Cn1;n2;:::;nk;(93)
we get
w./
nDn
X
s2S¤
n
sn.s/n
n1; n2; : : : ; nkk
Y
`D1
n`Sn`1./
Dn
X
n1Cn2CCnkDn;
2n1n2nkX
s2Cn1;n2;:::;nk
sn.s/n
n1; n2; : : : ; nkk
Y
`D1
n`Sn`1./:
Dn
X
n1Cn2CCnkDn;
2n1n2nk
n
n1; n2; : : : ; nkk
Y
`D1
n`Sn`1./X
s2Cn1;n2;:::;nk
sn.s/:
(94)
Now, Theorem 13 shows that w./
nD0for odd n, and that for even n,nD2m,
w./
2m D.1/m2m
X
m1Cm2CCmkDm;
1m1m2mk
2m
2m1; 2m2; : : : ; 2mk 2m
2m1; 2m2; : : : ; 2mk
k
Y
`D1
2m`A2m`1S2m`1./:
(95)
Using Lemma 14, we obtain the following theorem:
Theorem 18. For odd n,w./
nD0. For even n,nD2m > 0, we have
w./
2m D.1/m.2m/Š2X2mk
k
C
k1Šk2ŠkrŠ
k
Y
jD1
A2mj1
2mj.2mj1/Š2S2mj1.C=/;
where the sum runs over all partitions m1Cm2C C mkDmwith 1m1
m2 mk, and where k1,k2, ..., krare the corresponding multiplicities of
the different elements in the multi-set ¹m1; m2; : : : ; mkº:
464 R. Hudson, U. Schauz, and Y. Wu
From this and equation (54), we derive our final result:
Theorem 19. The nonzero moments of the quantum Lévy area y
B./
Œa:b/ are
EŒ.y
B./
Œa:b//2m
D.2m/Š.b a/2m X2mk
k
C
k1Šk2ŠkrŠ
k
Y
jD1
A2mj1
2mj.2mj1/Š2S2mj1.C=/;
(96)
where the sum runs over all partitions m1Cm2C C mkDmwith 1m1
m2 mk, and where k1,k2, ..., krare the corresponding multiplicities of
the different elements in the multi-set ¹m1; m2; : : : ; mkº:The Anare Euler zigzag
numbers and the Snare Euler polynomials.
6. The classical limit
We calculate the limit of EŒ.y
B./
Œa:b//2mas ! 1;or equivalently, C!1
2;
!1
2:Putting CDD1
2in (96), we get
lim
!1
EŒ.y
B./
Œa:b//2m
D.2m/Šba
22m X1
k1Šk2ŠkrŠ
k
Y
jD1
A2mj1
2mj.2mj1/Š2S2mj1.1/;
(97)
where the sum runs over all partitions m1Cm2C C mkDm, and where k1,
k2, ..., krare the corresponding multiplicities of the elements of the multi-set
¹m1; m2; : : : ; mkº. But, by the definition of the Euler polynomial,
Sn.1/ WD X
s2Sn
1des.s.1/;s.2/;:::;s.n// D jSnj D nŠ: (98)
So, using Lemma 10 and Lemma 14, we see that
lim
!1
EŒ.y
B./
Œa:b//2mD.2m/Šba
22m X1
k1Šk2ŠkrŠ
k
Y
jD1
A2mj1
.2mj/Š
Dba
22m
A2m:
(99)
This result is in agreement with the main theorem in [11].
Moments of quantum Lévy areas 465
References
[1] N. Bourbaki, Elements of mathematics. Algebra, Part I: Chapters 1-3. Translated from
the French. Hermann, Paris, and Addison-Wesley, Reading, MA, 1974. MR 0354207
Zbl 0281.00006
[2] P. B. Cohen, T. M. W. Eyre, and R. L.bHudson, Higher order Itô product formula
and generators of evolutions and flows. Internat. J. Theoret. Phys. 34 (1995), no. 8,
1481–1486. Proceedings of the International Quantum Structures Association, Quan-
tum Structures ’94 (Prague, 1994). MR 1353690 Zbl 0850.81035
[3] S.Chen and R. L. Hudson, Some properties of quantum Lévy area in Fock and non-
Fock quantum stochastic calculus. Probab. Math. Statist. 33 (2013), no. 2, 425–434.
MR 3158567 Zbl 1283.81099
[4] D. Foata and M. Schützenberger, Théorie géométrique des polynómes eulériens.
Lecture Notes in Mathematics, 138. Springer-Verlag, Berlin etc., 1970. MR 0272642
Zbl 0214.26202
[5] M. Hoffman, The algebra of multiple harmonic series. J. Algebra 194 (1997), no. 2,
477–495. MR 1467164 Zbl 0881.11067
[6] R. L.Hudson, Sticky shuffle Hopf algebras and their stochastic representations. In
H. Zhao and A. Truman (eds.), New trends in stochastic analysis and related topics.
A volume in honour of Professor K. D. Elworthy. World Scientific, Hackensack, N.J.,
2012, 165–181. MR 2920199 Zbl 1266.81123
[7] R. L. Hudson, Quantum Lévy area as quantum martingale limit. In L. Accardi
and F. Fagnola (eds.), Quantum probability and related topics. Proceedings of the
32nd Conference held in Levico Terme, May 29–June 4, 2011. Quantum Probabil-
ity and White Noise Analysis, 29. World Scientific, Hackensack, N.J., 2013, 169–188.
MR 3288045 Zbl 1327.81253
[8] R. L. Hudson and J. M. Lindsay, A non-commutative martingale representation
theorem for non-Fock quantum Brownian motion. J. Funct. Anal. 61 (1985), no. 2,
202–221. MR 0786622 Zbl 0577.60055
[9] R. L. Hudson, U.Schauz, and Y.Wu, The moments of Lévy’s area using a sticky shuffle
Hopf algebra. Commun. Stoch. Anal. 11 (2017), no. 3, 287–299. MR 3734126
[10] R. L. Hudson and Y. Pei, On a causal quantum stochastic double product integral
related to Lévy area. To appear in Ann. Inst. Henri Poincaré Comb. Phys. Interact.
[11] D. Levin and M. Wildon, A combinatorial method of calculating the moments of
Lévy area. Trans. Amer. Math. Soc. 360 (2008), no. 12, 6695–6709. MR 2434307
Zbl 1152.60064
[12] P. Lévy, Le mouvement Brownien plan. Amer. J. Math. 62 (1940). 487–550.
MR 0002734 Zbl 0024.13906
466 R. Hudson, U. Schauz, and Y. Wu
[13] P. Lévy, Wiener’s random function and other Laplacian functions. In J. Neyman (ed.),
Proceedings of the second Berkeley symposium on mathematical statistics and prob-
ability. Held at the Statistical Laboratory, Department of Mathematics, University of
California, July 31-August 12, 1950. University of California Press, Berkeley and Los
Angeles, 1951, 171–187. MR 0044774 Zbl 0044.13802
[14] K. Newman and D. E. Radford, The cofree irreducible Hopf algebra on an algebra.
Amer. J. Math. 101 (1979), no. 5, 1025–1045. MR 0546301 Zbl 0422.16003
[15] K. R. Parthasarathy, An introduction to quantum stochastic calculus. Monographs in
Mathematics, 85. Birkhäuser Verlag, Basel, 1992. MR 1164866 Zbl 0751.60046
[16] T. K. Petersen, Eulerian numbers. With a foreword by R. Stanley. Birkhäuser Ad-
vanced Texts: Basler Lehrbücher. Birkhäuser/Springer, New York, 2015. MR 3408615
Zbl 1337.05001
[17] R. P. Stanley, A survey of alternating permutations. In R. A. Brualdi, S. Hedayat,
H. Kharaghani, G. B. Khosrovshahi, and S. Shahriari (eds.), Combinatorics and
graphs. Selected papers from the 20th International Anniversary Conference (IPM
20) held in Tehran, May 15–21, 2009. Contemporary Mathematics, 531. American
Mathematical Society, Providence, R.I., 2010, 165–196. MR 2757798 Zbl 1231.05288
©European Mathematical Society
Communicated by Frédéric Patras
Received January 30, 2017; revised May 14, 2017; accepted June 10, 2017
Robin Hudson, Department of Mathematical Sciences, Loughborough University,
LE11 3TU,UK
e-mail: [email protected]
Uwe Schauz, Department of Mathematical Sciences,
Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China
e-mail: uwe.schauz@xjtlu.edu.cn
Yue Wu, Technische Universität Berlin, Institut für Mathematik, Secr. MA 5-3,
Straße des 17. Juni 136, DE-10623 Berlin, Germany
e-mail: [email protected]lin.de
School of Engineering, University of Edinburgh, EH9 3JL, UK
e-mail: [email protected]c.uk
