Character expansion of matrix integrals

Character expansion of matrix integrals J.W. van de Leur∗

A. Yu. Orlov†

November 15, 2016

arXiv:1611.04577v1 [nlin.SI] 14 Nov 2016

Abstract We consider expansions of certain multiple integrals and BKP tau functions in characters of orhtogonal and symplectic groups. In particular we consider character expansions of integrals over orthogonal and over symplectic matrices.

Key words: matrix integrals, β = 2 ensembles, integrable systems, tau functions, Pfaff lattice, BKP, DKP, O(N ) and Sp(N ) characters, free fermions

1

Introduction

Some notations Let us recall that the characters of the unitary group U(n) are labeled by partitions and coincide with the so-called Schur functions [19]. A partition λ = (λ1 , . . . , λn ) is a set of nonnegative integers λi which are called parts of λ and which are ordered as λi ≥ λi+1 . The number of non-vanishing P parts of λ is called the length of the partition λ, and will be denoted by ℓ(λ). The number |λ| = i λi is called the weight of λ. The set of all partitions will be denoted by P. The Schur function corresponding to λ is defined as the following symmetric function in variables x = (x1 , . . . , xn ) : i h det xλj i −i+n   i,j sλ (x) = det x−i+n j i,j

(1)

in case ℓ(λ) ≤ n and vanishes otherwise. One can see that sλ (x) is a symmetric homogeneous polynomial of degree |λ| in the variables x1 , . . . , xn . Remark 1. In case the set x is the set of eigenvalues of a matrix X, we also write sλ (X) instead of sλ (x). There is a different definition of the Schur function as quasi-homogeneous non-symmetric polynomial of degree |λ| in other variables, p = (p1 , p2 , . . .), where deg pm = m:   (2) sλ (p) = det s(λi −i+j) (p) i,j

P P m 1 and the Schur functions s(i) are defined by e m>0 m pm z = m≥0 s(i) (p)z i . The Schur functions defined by (1) and by (2) are equal, sλ (p) = sλ (x), provided the variables p and x are related by X xm (3) pm = i i

From now on, we will use in case the argument of sλ is written as a fat letter the definition (2), and we imply the definition (1) otherwise. Remark 2. For functions f (p) = f (p(A)), where pm (A) := Tr Am , m = 1, 2, . . . and A is a given matrix we may equally write either f (p(A)) or f (A) where the capital letter implies a matrix. In particular under this convention we may write sλ (A) and τ (A) instead of sλ (p(A)) and τ (p(A)). ∗ Mathematical Institute, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands, email: [email protected] † Institute of Oceanology RAS, Nahimovskii Prospekt 36, Moscow, Russia,, and National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia, email: [email protected]

1

Integrals over the unitary group. Consider the following integral over the unitary group which depends on two semi-infinite sets of variables p = (p1 , p2 , . . .) and p∗ = (p∗1 , p∗2 , . . .), which are free parameters Z ∗ −1 I (p, p∗ ) := etrV (p,U)+trV (p ,U ) d U = (4) ∗

U(n)

U(n)

1 (2π)n

Z

Y

0≤θ1 ≤...≤θn ≤2π 1≤j0 m

(pm eimθj +p∗m e−imθj ) dθj

(5)

j=1

X1 pn xn n n>0

(6)

Here d∗ U is the Haar measure of the group U(n), see (177) in Appendix, and eiθ1 , . . . , eiθn are the eigenvalues of U ∈ U(n). The exponential factors inside the integral may be treated as a perturbation of the Haar measure and parameters p, p∗ are called coupling constants. Using the Cauchy-Littlewood identity X P∞ 1 ∗ τ (p|p∗ ) := e m=1 m pm pm = sλ (p∗ )sλ (p) (7) λ∈P

and the orthogonality of the irreducible characters of the unitary group Z sλ (U )sµ (U −1 )d∗ U = δλ,µ we obtain that IU(n) (p, p∗ ) =

X

(8)

sλ (p)sλ (p∗ )

(9)

λ∈P ℓ(λ)≤n

which express the integral over unitary matrices as the ”perturbation series in coupling constants”. The formula (9) first appeared in [21] in the context of the study of Brezin-Gross-Witten model. It was shown there that the integral IU(n) (p, p∗ ) may be related to the Toda lattice tau function of [10] and [32] under certain restriction. Then, the series in the Schur functions (9) may be related to the double Schur functions series found in [30] and [31]. In this paper we want to express integrals over the symplectic and over the orthogonal groups, ISp(N ) (p) and IO(N ) (p) respectively, as sums of product of characters of the orthogonal and of symplectic groups, i.e. to obtain the analogues of the relation (9) and relate these integrals and sums to integrable systems. On the one hand we shall relate ISp(N ) (p) and IO(2n) (p) to the DKP1 , and we shall relate IO(2n+1) (p) to BKP tau functions, introduced respectively in [10] and [11] and obtain Pfaffian representation for these integrals. On the other hand one can relate these integrals to the Toda lattice (TL) tau function [10], [32] which yields the determinant representation. We show that the so-called β = 1, 2, 4 ensembles may be written as formal series in characters.

Polynomials oλ (p) and spλ (p) and TL tau functions τ±(p|p∗)

2

The orthogonal and symplectic characters are also labeled by partitions. They are given by the following expressions i h det xjλi +n−i+1 − xj−λi −n+i   1≤i,j≤n (10) oλ = −n+i det xn−i+1 − x j j 1≤i,j≤n

and

1 We

i h i −n+i−1 det xjλi +n−i+1 − x−λ j  n−i+1  1≤i,j≤n spλ = det xj − xj−n+i−1 1≤i,j≤n

(11)

need to note that the DKP hierarchy has other names. It was rediscovered in [2] using the approach different of [10] and called Pfaff lattice. It was also called coupled KP equation in [9]

2

respectively. See [6] or Appendix A.5 for more information. Baker [3] realized that these characters can be obtained from the corresponding Schur functions sλ by action of some operator. For this, it will be convenient to use the Schur functions in terms of the power sums pm . As usual, write ∂˜ = (∂p1 , 2∂p2 , 3∂p3 , . . .). Let   X 1 X m 1 ˜ := − − (∂m )2 ∓ ∂2m Ω∓ (p) = (12) p2m ∓ p2m , Ω∓ = Ω∓ (∂) 2m 2m 2 m>0 m>0 then

˜

˜

oλ (p) = eΩ− (∂) · sλ (p) ,

spλ (p) = eΩ+ (∂) · sλ (p)

˜∗

(13)

Hence, if we let the operator Ω∓ (∂ ) act on the Cauchy- Littelewood identity (7), we obtain X τ− (p|p∗ ) = oλ (p∗ )sλ (p)

(14)

λ

and τ+ (p|p∗ ) =

X

spλ (p∗ )sλ (p)

(15)

λ

where

1

τ∓ (p|p∗ ) = e− 2 Remark 3. Note that ∗

P∞

1 2 m=1 m pm

∓

P∞

1 m=1 2m p2m +

P∞

∗ 1 m=1 m pm pm

(16)

τ∓ (p|p∗ ) = eΩ∓ (p) τ0 (p|p∗ )

(17)

∗ 1 m>1 m pm pm

P

where τ0 (p|p ) = e is known to be the simplest tau function of the TL hierarchy (this simplest tau function does not depend on the discrete TL time p0 ).

It is well known that the function τ0 (p|p∗ ), for the variables p = (p1 , p2 , . . .), is a solution of the Hirota bilinear equations for the KP hierarchy: I dz V (p′ −p,z) e τ0 (p′ − [z −1 ]|p∗ )τ0 (p + [z −1 ]|p∗ ) = 0 (18) 2πi Here V is given by (6) and the variables p∗ = (p∗1 , p∗2 , . . .) play the role of auxiliary parameters. Here
and below the notation [a] serves to denote the following set of power sums: a, a2 , a3 , . . . . The action ˜∗ ˜′∗ of eΩ∓ (∂ ) eΩ∓ (∂ ) on (18) gives I dz V (p′ −p,z) e τ∓ (p′ − [z −1 ]|p∗ )τ∓ (p + [z −1 ]|p∗ ) = 0 (19) 2πi hence τ∓ (p|p∗ ) is also a tau function of the KP hierarchy. Then it follows from Remark 3 that both τ± (p|p∗ ) are TL tau functions where p and p∗ are two sets of the higher times. These tau functions do not depend on the discrete TL variable t0 because τ0 (p|p∗ ) does not depend on it. According to Sato [26] a KP tau function may be related to an element of an infinite dimensional Grassmannian as a series in the Schur functions X tauKP (p) = πλ sλ (p) λ

where πλ are the Pl¨ ucker coordinates of the element. Hence according to (7), (14) and (15), the functions sλ (p∗ ), oλ (p∗ ) and spλ (p∗ ) are the Pl¨ ucker coordinates of τ0 (p|p∗ ), τ− (p|p∗ ) and τ (p|p∗ ), respectively. The related elements of the Grassmannian is written down in Appendix A.9. The Pl¨ ucker coordinates oλ (p∗ ) and spλ (p∗ ) may be evaluated respectively as follows   P P∞ P∞ ∗ 1 2 1 1 ˜ · e− 12 ∞ m=1 m pm − m=1 2m p2m + m=1 m pm pm oλ (p∗ ) = sλ (∂) |p=0 (20) and

 P ˜ · e− 21 ∞ m=1 spλ (p∗ ) = sλ (∂)

1 2 m pm

+

3

P∞

1 m=1 2m p2m +

P∞

∗ 1 m=1 m pm pm



|p=0

(21)

which may be compared with the identity for the Schur functions   P ∗ 1 ˜ ·e ∞ m=1 m pm pm sλ (p∗ ) = sλ (∂) |p=0

(22)

As in the previous section, let us assign the weight k to pk . We recall that the polynomials sλ are quasi-homogeneous in the variables pm of the weight |λ|. As we see from (20) and (21) polynomials oλ and spλ are not quasi-homogeneous: they both may be presented as sλ plus polynomials of minor weights. For instance o(1) (p) = sp(1) (p) = s(1) (p) = p1 1 1 1 1 p2 + p21 − 1 , sp(2) (p) = s(2) (p) = p2 + p21 2 2 2 2 1 2 1 2 1 1 o12 (p) = s12 (p) = − p2 + p1 , sp12 (p) = s12 (p) − 1 = − p2 + p1 − 1 2 2 2 2 o(2) (p) = s(2) (p) − 1 =

Next from τ− (−p| − p∗ ) = τ+ (p|p∗ ) and from sλ (p) = (−)|λ| sλtr (−p), we get spλ (p) = (−)|λ| oλtr (−p)

(23)

where −p = (−p1 , −p2 , −p3 , . . .). From the following well-known formulas (see [19], pages 76 and 77 or [18] page 238) X P P X X 1 1 2 1 e 2 m>0 m pm + m>0,odd m pm = sµ (p) , e−Ω+ (p) = sµ (p) sµ∪µ (p) , and e−Ω− (p) = µ∈P

µ∈P

µ∈Peven

where Peven is the set of all partitions with even parts (including (0)), one deduces Lemma 1. 1

˜ z |µ| sµ (∂)

=

e2

˜ z 2|µ| sµ∪µ (∂)

=

e2

˜ z |µ| sµ (∂)

=

e2

X

P∞

2 mz 2m ∂m +

P

P∞

2 mz 2m ∂m −

P

P∞

2 mz 2m ∂m +

P

m=1

m>0,odd

z m ∂m

(24)

µ∈P

X

1

m=1

m>0, even

z m ∂m

= e−Ω+ (z

m˜ ∂m )

z m ∂m

= e−Ω− (z

m

m>0, even

(25)

µ∈P

X

1

m=1

∂˜m )

(26)

µ∈Peven

˜ is defined as in (20)-(21). where sλ (∂) (see also [22] where in (25) and (26) there is the opposite sign for linear term in exponents which is a misprint). Remark 4. From Lemma 1 a number of relations may be obtained. We present two examples: e−

P 2 1 1 m>0 2m pm − m>0, odd m pm

P

X

sµ/λ (p) =

X

sλ/µ (p) = e

P∞

2 m=1 (m∂m + ∂m )

· oλ (p)

µ∈P

µ∈P

where the first equality is obtained from Ex 27(a) in I.5 of [19]. The second example follows from (13), (24) and ˜ · sλ (p) = sλ/µ (p). The second example, from sµ (∂) X

sλ/µ∪µ (p) = e

P

2 m>1 m∂m

· oλ (p)

(27)

µ∈P

h P i 2 we obtain from (13) and (25). Note, that the constant term e m>1 m∂m · oλ (p)

p=0

λ of form µ ∪ µ, and vanishes otherwise.

4

of (27) is equal to 1 for any

Relation to irreducible characters of the orthogonal and symplectic groups. We shall use notations explained in Remark 2 with pm (U ) = Tr U m . In this notation we write τ+ (U |p∗ ) = τ− (U |p∗ ) =

n Y X Y P∞ m 1 ∗ (1 − xi xj ) e m=1 m pm xk = spλ (p∗ )sλ (U ) i 0 0 n−1 hn| = , |ni = (36) † † ψ if n < 0 h0|ψ−1 · · · ψ−n if n < 0 −n · · · ψ−1 |0i then hn| · 1 · |mi = δn,m . Note that eΩ∓ is an automorphism of the Fock space. It maps any charge sector into itself and maps Schur functions into orthogonal and symplectic characters, see (13). Thus oλ (for eΩ− ) and spλ (for eΩ+ ) satisfies ˜ ˜ ˜ ˜ Resz eΩ∓ (α) ψ(z)e−Ω∓ (α) σ ⊗ eΩ∓ (α) ψ † (z)e−Ω∓ (α) σ = 0,

(37)

˜ ˜ ˜ ˜ ψ(z)e−Ω∓ (α) and Ψ†∓ (z) = eΩ∓ (α) ψ † (z)e−Ω∓ (α) . where σ = eΩ∓ τ . We now want to calculate Ψ∓ (z) = eΩ∓ (α) † We use the vertex operator expression for ψ(z) and ψ (z)

ψ(z) = eα0 z α0 e−

P

ψ † (z) = e−α0 z −α0 e where αm =

X

αi i0 z i

P

e−

αi i0 z i

P

(39)

† : ψi ψi+m :

(40)

i∈Z

for future use we also introduce Γ(p) := e

P∞

m=1 tm αm

Γ† (p) := e

,

P∞

m=1 tm α−m

(41)

Note that [αi , αj ] = iδi,−j , hence they form a Heisenberg algebra. Now use the standard realization of the Heisenberg algebra αk = ∂k , α−k = ktk , α0 = q∂q , eα0 = q Then (38), respectively (39) turn into ψ(z) = qz q∂q e

P∞

i=1 ti z

ψ † (z) = q −1 z −q∂q e−

i

e−

P∞

P∞

i=1

i=1 ti z

i

e

−i

∂i z i

P∞

i=1

(42) −i

∂i z i

(43)

Using the following formulas which can easily be deduced from the Cambell-Baker-Hausdorff formula: 2

ea∂x ebx = ebx ea(b+∂x ) ,

ea∂x ebx = ebx ea(b+∂x )

one thus obtains: 1

1

Ψ∓ (z) = (1 − z 2 ) 2 ± 2 qz q∂q e Ψ†∓ (z) = (1 − z 2 )

1 1 2∓2

P∞

i=1 ti z

q −1 z −q∂q e−

P∞

i

P∞

e−

i=1

i=1 ti z

i

e

∂i z

2

−i +z i i

z −i +z i i=1 ∂i i

P∞

(44) (45)

Hence the orthogonal and symplectic characters satisfy the bilinear equation: Resz Ψ∓ (z)σ∓ ⊗ Ψ†∓ (z)σ∓ = 0,

(46)

or equivalently Resz (1 − z 2 )e

P∞

i=1 (ti −si )z

i

σ∓ (t − [z] − [z −1 ])σ∓ (s + [z] + [z −1 ]) = 0,

(47)

which is equation (5.4) of Baker [3]. P P Now note that if we write Ψ∓ (z) = i∈Z Ψ∓i z i and Ψ†∓ (z) = i∈Z Ψ†∓i z −i−1 , then the modes still satisfy the usual relations. Ψ∓i Ψ∓j + Ψ∓j Ψ∓i = 0 = Ψ†∓i Ψ†∓j + Ψ†∓j Ψ†∓i

Ψ∓i Ψ†∓j + Ψ†∓j Ψ∓i = δij

Using the above vertex operators on the vacuum |0i = q 0 one still has Ψ∓i |0i = 0 = Ψ†∓(−i−1) |0i 6

i1 ti αi

g|0i

for g ∈ Gl∞

changes into σ∓ (p) = eΩ∓ τ (p), which is equal to σ∓ (p) = h0|e

P

2 1 1 i>1 ti αi − 2i αi ∓ 2i α2i

g|0i

for g ∈ Gl∞

which corresponds to the modified Hamiltonian of [3], Section 3, Approach I. Next calculate h0|e

2 1 1 i>1 ti αi − 2i αi ∓ 2i α2i

P

e

∗ i>1 ti α−i

P

|0i = τ∓ (p∗ |p)

Hence, it makes sense to look at h0|e

2 1 1 i>1 ti αi − 2i αi ∓ 2i α2i

P

ge

∗ i>1 ti α−i

P

7

|0i

for g ∈ Gl∞

Remark 5. Actually we have

˜

τ (p, p∗ ) → τ ± (p, p∗ ) = eΩ± (∂) · τ (p, p∗ ) If

τ (p, p∗ ) =

X

(50)

sλ (p)πλ,µ sµ (p∗ )

λ,µ∈P

where

πλ,µ = h0| sλ (α) ˜ g s µ (α ˜ ∗ ) |0i

then τ ± (p, p∗ ) =

X

± sλ (p)πλ,µ sµ (p∗ ) ,

λ,µ∈P

where

+ πλ,µ = h0| spλ (α) ˜ g s µ (α ˜ ∗ ) |0i ,

Similarly, one can consider τ

4

a,b

− πλ,µ = h0| oλ (α) ˜ g s µ (α ˜ ∗ ) |0i

with a, b = ±.

Integrals over symplectic group and over orthogonal groups

Haar measures and generating functions for characters. Lemma 6 in Appendix A.1 and formulae of the Appendix A.8 results in the following lemmas we shall need: Lemma 2. The Haar measures of the symplectic group Sp(2n) and of the unitary group U(n) are related as follows etrV (S,p) d∗ S

2−n τ− (U |p)τ− (U −1 |p)d∗ U

=

−n

= =

−1

(51) 2

2 τ+ (U |p)τ+ (U |p)det(1 − U )det(1 − U 2−n τ+ (U |p)τ− (U −1 |p)det(1 − U 2 )d∗ U

−2

)d∗ U

(52) (53)

where eiθ1 , e−iθ1 , . . . , eiθn , e−iθn are eigenvalues of S ∈ Sp(2n) while eiθ1 , . . . , eiθn are eigenvalues of U ∈ U(n). Lemma 3. The Haar measures of the orthogonal group O(2n) and of the unitary group U(n) are related as follows etrV (O,p) d∗ O

= = =

2−n τ+ (U |p)τ+ (U −1 |p)d∗ U 2−n τ− (U |p)τ− (U −1 |p)det(1 − U 2 )−1 det(1 − U −2 )−1 d∗ U

2−n τ− (U |p)τ+ (U −1 |p)det(1 − U 2 )−1 d∗ U

(54) (55) (56)

Lemma 4. The Haar measures of the orthogonal group O(2n + 1) and of the unitary group U(n) are related as follows etrV (O,p) d∗ O

= = =

2−n τ+ (U |p)τ+ (U −1 |p)det(1 − U )det(1 − U −1 )d∗ U 2

−n

τ− (U |p)τ− (U

−1

−1

|p)det(1 + U ) det(1 + U 1 − U −1 2−n τ− (U |p)τ+ (U −1 |p)det d∗ U 1+U

−1 −1

)

(57) d∗ U

(58) (59)

where eiθ1 , e−iθ1 , . . . , eiθn , e−iθn , 1 are eigenvalues of O ∈ O(2n + 1) while eiθ1 , . . . , eiθn are eigenvalues of U ∈ U(n).

4.1

Integrals over symplectic group.

Consider the following integral over the symplectic group Z P∞ m ISp(2n) (p) = e m=1 tm trS d∗ S

(60)

S∈Sp(2n)

where d∗ S is the corresponding Haar measure. Explicitly 2 Z n n Y Y P∞ 2n e2 m=1 tm cos mθi sin2 θi dθi ISp(2n) (p) = n (cos θi − cos θj )2 π 0≤θ1 ≤···≤θn ≤π ij Thus each relation of Proposition 6 is proven. 12

Pfaffian representation. Let us mark that thanks to the Wick’s rule applied to calculate (92) we directly obtain the Pfaffian representation of the integral (83) as follows Proposition 7. IO(2n) (p) = Pf [Mkj (p)]k,j=1,...,2n+2

(100)

where in case N even 1 Mkj (p) = 4πi

P∞
m −m dx 1 xj−k − xk−j (x − x−1 )−1 e m=1 m pm (x +x ) x

I

(101)

In case N = 2n + 1 the entries Mkj k, j ≤ 2n + 1 as before and I 1 Mk,2n+2 = −M2n+2,k = xk eV (x,p) dν(x) 2

(102)

For the proof we notice that from Wick’s rule we get Z ψ(x−1 )ψ(x) † † dµ(z(x))|0i Mkj (p) = h0|ψk ψj x−1 − x Relation of the integral over O(N ) to an integral over U(2n). Next turn to its two-component KP counterpart. According to Proposition 4 in [17] we have

hN, −N |e

P∞

1 m=1 m pm

1 (2) (α(1) m −αm ) e 4πi

H


2 IO(N ) (p) =

(1) 1 √ (1)ψ † −ψψ †(2) (1)) ψ (1) (x−1 )ψ †(2) (x)(x−x−1 )−1 dx x + 2 (ψ

(a)

For the two-component fermions see Appendix, αn :=

P

(a)

i∈Z

†(a)

|0, 0i

(103)

ψi ψi+n .

˜ = (˜ As 1D TL-NLS tau function. Denote z = x + x−1 and introduce variables p p0 , p˜1 , . . .) with the help of (78). Then similar to (79) we get I(p(˜ p)) =

Z

2

−2

5

···

Z

2

n P Y Y ∞ m 1 e m=1 m p˜m zi +p˜0 dzi (zi − zj )2

−2 i1 and variables T = (T1 , T2 , . . .) and p = (p1 , p2 , . . .) are linearly dependent and related as follows V (z, T ) = V (x, p) + V (x−1 , p) + c(p), For instance, T1 = t1 − Then we have

P∞

n=1 (2n

c(p) =

∞ X (2n)! t2n n!n! n=1

(106)

+ 1)t2n+1 .

Proposition 8. In (T ) =

Z

∆(z1 , . . . , zn )2

n Y

i=1

13

ziN eV (zi ,T ) dµ(zi ) =

(107)

=

X

h1 >···>hn ≥0

=

00 (2) s{h} (p(1) ) π{h, ) ˜ (p ˜ (N ) s{h} h}

(109)

X

−− c(p) ec(p) o{h} (p) π{h, ˜ (p) e ˜ (N ) o{h} h}

(110)

X

++ c(p) ec(p) sp{h} (p) π{h, ˜ (p) e ˜ (N ) sp{h} h}

(111)

X

−+ c(p) ec(p) o{h} (p) π{h, ˜ (p) e ˜ (N )sp{h} h}

(112)

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

=

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

=

(108)

X

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

=

s{h} (T ) π{h} (N )

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

(1)

(2)

where T = p(1) − p(2) (that is Tm = tm − tm ), and i h 00 (N ) π{h} (N ) = n!det πn−i,h j

(113)

i,j=1,...,n

i h ab ab π{h, (N ) = n!det π ˜ ˜ h} h ,h i

j

i,j=1,...,n

,

a, b = ±, 0

(114)

and the related moment matrices are

00 πi,j (N )

=

++ πi,j (N )

=

Z

Z

z i+j+N dµ(z)

(115)

xi−j (x + x−1 )N dµ(z(x))

(116)

dµ(z(x)) (1 − x2 )2 Z dµ(z(x)) = xi−j (x + x−1 )N 1 − x2

−− πi,j (N )

=

−+ πi,j (N )

Z

xi−j (x + x−1 )N

(117) (118)

Proof. We use Lemma 6 in the Appendix A.1 and relations (106), (28)-(29). The Schur functions which are involved in (28)-(29) we present as ratios of the determinants (1), at last we use the simple identity (sometimes called Andreif identity) Z

det [τi (xj )]i,j=1,...,n det [τi (xj )]i,j=1,...,n

n Y

dµ(xi ) = n!det

i=1

Z

 τi (x)τj (x)dµ(x)

i,j=1,...,n

to get moment matrices (115)-(118). Remark 7. Series (108) is a special case of (109) where p(2) = 0. β = 2 ensembles as the BKP tau function. In (T (p)) = n!ec(p) hn|e

P

Using (147) in Appendix one may verify that

1 m>0 m pm αm

e

R

√ R ψ(x−1 )ψ(x) dµ(z(x))+ 2 x−1 −x

ψ(x)dµ(x)φ

|0i

(119)

Then we can apple the Wick’s rule to get the Pfaffian representation In (T (p)) = n! Pf [Aij (p)]

(120)

where, for n even Remark 8. The integrals (107) may be related both to the BKP hierarchy where p are BKP higher times the ˜ , see (78) one-dimensional Toda chain with higher times p

14

6

The character expansion of two-matrix models

Let z = x + x−1 ,

z˜ = x ˜+x ˜−1

(121)

˜ In case we can express the variables p as a linear combination of the variables p(1) and the variables p as a linear combination of the variables p(2) : tm =

∞ X

Dmn t(1) n ,

t˜m =

n=1

∞ X

˜ mn t(2) D n

n=1

in such a way that V



az + b (1) ,p cz + d



= V (x, p) ,

V

a ˜z˜ + ˜b (1) ,p c˜z˜ + d˜

!

˜) = V (x, p

(122)

we can apply the same method based on (140)-(141) in Appendix A.1 to get character expansion of Z

In (p(1) , p(2) ) =

∆(z1 , . . . , zn )∆(˜ z1 , . . . , z˜n )

n Y

ziN eV (˜zi ,p

(2)

)−V (˜ zi ,p(2) )

dµ(zi , z˜i )

(123)

i=1

For the sake of simplicity let us take dµ(z, z˜) = 0 if either z ≥ 1 or z˜ ≥ 1, and consider V (z −1 , p(1) ) = V (x, p) , (1)

(1)

˜) V (¯ z −1 , p(1) ) = V (x, p

(1)

(1)

such that t1 = t1 , t2 = t2 , t3 = t3 − t(1)1 , t4 = t4 − 2t(1)2 and so on. In (p(1) , p(2) ) = =

X

(2) ) s{h} (p(1) ) G00 ˜ (p ˜ s{h} {h,h}

(125)

X

˜ ) o{h} ec(p) o{h} (p) G−− p) ec(˜p) ˜ (˜ ˜ (p, p {h,h}

(126)

X

˜ ) sp{h} ec(p) sp{h} (p) G++ p) ec(˜p) ˜ (˜ ˜ (p, p {h,h}

(127)

X

˜ ) sp{h} ec(p) o{h} (p) G−+ p) ec(˜p) ˜ (˜ ˜ (p, p {h,h}

(128)

X

s{h} (p(1) ) G0+ p) sp{h} p) ec(˜p) ˜ (˜ ˜ (˜ {h,h}

(129)

X

s{h} (p(1) ) G0− p) o{h} p) ec(˜p) ˜ (˜ ˜ (˜ {h,h}

(130)

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

=

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

=

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

=

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

=

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

=

h1 >···>hn ≥0 ˜ >···>h ˜ n ≥0 h 1

where

(124)

i h ab Gab = det G ˜ ˜j {h,h} hi , h

i,j=1,...,n

,

a, b = ±, 0

and the related moment matrices are Z G00 = z i+N z˜j+N dµ(z, z˜) i,j Z n(n−1) n(n−1) ++ Gi,j = xi− 2 x ˜j− 2 (x + x−1 )N (˜ x+x ˜−1 )N dµ(z(x), z˜(˜ x)) Z n(n−1) n(n−1) dµ(z(x), z˜(˜ x)) ˜j− 2 (x + x−1 )N (˜ x+x ˜−1 )N G−− = xi− 2 x i,j (1 − x2 )(1 − x ˜2 ) 15

(131)

(132) (133) (134)

G−+ i,j G0+ i,j G0− i,j

= = =

Z

Z

Z

xi−

n(n−1) 2

x ˜j−

n(n−1) 2

(x + x−1 )N (˜ x+x ˜−1 )N

z i+N x ˜j−

n(n−1) 2

p(1) (˜ x+x ˜−1 )N dµ(z, z˜(˜ x))

z i+N x ˜j−

n(n−1) 2

(˜ x+x ˜−1 )N

dµ(z(x), z˜(˜ x)) 2 1−x

dµ(z, z˜(˜ x)) 1−x ˜2

(135) (136) (137)

dzd˜ z Example. Take z˜ = z¯, x = x ¯, t(1) = t¯(2) , tm = t¯m , and N = 0. Take also dµ(z, z˜) = f (|z|) |1−x 2 |2 , where z and x are related by (121). Then X In (p(1) , p(2) ) = rλ spλ (p)sp(¯ p) (138) λ

Acknowledgements A.O. was supported by RFBR grant 14-01-00860. This work has been funded by the Russian Academic Excellence Project ’5-100’.

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A A.1

Appendices Rewriting Vandermonde determinants

We have usefull elementary

17

Lemma 6. Let z = x + x−1 , (Joukowsky transform). Then

Y

1≤k