arXiv:math/0010135v1 [math.CO] 13 Oct 2000 n hi plctos ahmtclSine eerhInstitut Research Sciences Mathematical 2001. Press, : University Applications” Their and rnesUiest,Wlhm as044 S.Emi:vanmoerbe acknow gratefully Foundation is Science grant E-mail: Foundation National Francqui USA. a a and of FNRS 02454, a support Nato, Mass The Waltham, @math.brandeis.edu. University, Brandeis nerbeLtie:Rno arcsadRandom and Matrices Random Lattices: Integrable ∗ † xaddvrino etrsa SI ekly pig19.T a To 1998. Spring Berkeley, MSRI, at lectures of version Expanded eateto ahmtc,Uiested ovi,14 Louvain 1348 Louvain, de Universit´e Mathematics, of Department ieeta qain o h rgnlprobabilities. original the for equ equations integrable the differential determina with integ Fredholm combined for which or constraints, also integrals rasoro matrix but These lattices, Toeplitz int equation. and KdV matrix Pfaff these Toda, that the tau-functio A like showing natural are to spaces. time-parameters, devoted adding symmetric s priately is over symmetric lectures integrals to these as of space finally tangent and the groups over over integrals determinan Fredholm as T (or naturally integrals permutations. matrix random suggest and infinite) problems and (finite matrices dom hs etrspeetasre frcn eeomnsi th in developments recent of survey a present lectures These irevnMoerbeke van Pierre Permutations ac 5 2000 25, March Abstract 1 ∗ † sfritgal lattices, integrable for ns ledged. ulctos#0 Cambridge #40, Publications e tosla o(partial) to lead ations al D’,lk the like PDE’s, rable s,wihaievery arise which ts), rn M-84570 a DMS-98-4-50790, # grant pa n”admMatrices ”Random in ppear gas pnapro- upon egrals, t lostsyVi- satisfy also nts ae,a integrals as paces, motn part important n eeprobabilistic hese l-ev,Blimand Belgium -la-Neuve, ego.c.cb and [email protected] rao ran- of area e 0, p.2 § Contents

0 introduction 3

1 Matrix integrals, random matrices and permutations 5 1.1 Tangent space to symmetric spaces and associated random matrix en- sembles ...... 6 1.2 Infinite Hermitian matrix ensembles ...... 12 1.3 Integralsoverclassicalgroups ...... 15 1.4 Permutationsandintegralsovergroups ...... 17

2 Integrals, vertex operators and Virasoro relations 26 2.1 Beta-integrals ...... 26 2.1.1 Virasoro constraints for Beta-integrals ...... 26 2.1.2 Proof: β-integrals as fixed points of vertex operators ...... 29 2.1.3 Examples ...... 32 2.2 Doublematrixintegrals...... 34 2.3 Integralsovertheunitcircle ...... 36

3 Integrable systems and associated matrix integrals 38 3.1 Toda lattice and Hermitian matrix integrals ...... 38 3.1.1 Toda lattice, factorization of symmetric matrices and orthogonal polynomials ...... 38 3.1.2 SketchofProof ...... 41 3.2 Pfaff Lattice and symmetric/symplectic matrix integrals ...... 47 3.2.1 Pfaff lattice, factorization of skew-symmetric matrices and skew- orthogonalpolynomials...... 47 3.2.2 SketchofProof...... 51 3.3 2d-Toda lattice and coupled Hermitian matrix integrals ...... 56 3.3.1 2d-Toda lattice, factorization of moment matrices and bi-orthogonal polynomials ...... 56 3.3.2 Sketchofproof ...... 58 3.4 The Toeplitz Lattice and Unitary matrix integrals ...... 59 3.4.1 Toeplitz lattice, factorization of moment matrices and bi-orthogonal polynomials ...... 59 3.4.2 SketchofProof ...... 63

4 Ensembles of finite random matrices 64 4.1 PDE’s defined by the probabilities in Hermitian, symmetric and sym- plecticrandomensembles...... 64 4.1.1 Gaussian Hermitian, symmetric and symplectic ensembles . . . 66 4.1.2 Laguerre Hermitian, symmetric and symplectic ensembles . . . . 66 4.1.3 Jacobi Hermitian, symmetric and symplectic ensembles . . . . . 67

2 0, p.3 §

4.2 ODE’s, when E hasoneboundarypoint ...... 68 4.3 ProofofTheorems4.1,4.2and4.3 ...... 71 4.3.1 GaussianandLaguerreensembles ...... 71 4.3.2 Jacobiensemble...... 72 4.3.3 Inserting partials into the integrable equation ...... 73

5 Ensembles of infinite random matrices: Fredholm determinants, as τ-functions of the KdV equation 75

6 Coupled random Hermitian ensembles 84

7 Random permutations 85

8 Random involutions 89

9 Appendix: Chazy classes 91

0 introduction

The purpose of these lectures is to give a survey of recent interactions between statistical questions and integrable theory. Two types of questions will be tackled here: (i) Consider a random ensemble of matrices, with certain symmetry conditions to guarantee the reality of the spectrum and subjected to a given statistics. What is the probability that all its eigenvalues belong to a given subset E ? What happens, when the size of the matrices gets very large ? The probabilities here are functions of the boundary points ci of E. (ii) What is the statistics of the length of the largest increasing sequence in a random permutation, assuming each permutation is equally probable ? Here, one considers generating functions (over the size of the permutations) for the probability distributions, depending on the variable x. The main emphasis of these lectures is to show that integrable theory serves as a useful tool for finding equations satisfied by these functions of x, and conversely the probabilities point the way to new integrable systems. These questions are all related to integrals over spaces of matrices. Such spaces can be classical Lie groups or algebras, symmetric spaces or their tangent spaces. In infinite- dimensional situations, the ” -fold” integrals get replaced by Fredholm determinants. ∞ During the last decade, astonishing discoveries have been made in a variety of di- rections. A first striking feature is that these probabilities are all related to Painlev´e equations or interesting generalizations. In this way, new and unusual distributions have entered the statistical world. Another feature is that each of these problems is related to some integrable hier- archy. Indeed, by inserting an infinite set of time variables t1, t2, t3, ... in the integrals

3 0, p.4 §

∞ t yi or Fredholm determinants - e.g., by introducing appropriate exponentials e 1 i in P the integral - this probability, as a function of t1, t2, t3, ..., satisfies an integrable hier- archy. Korteweg-de Vries, KP, Toda lattice equations are only a few examples of such integrable equations. Typically integrable systems can be viewed as isospectral deformations of differential or difference operators . Perhaps, one of the most startling discoveries of integrable theory is that can beL expressed in terms of a single“τ-function” τ(t , t , ...) (or vector L 1 2 of τ-functions), which satisfy an infinite set of non-linear equations, encapsulated in a single “bilinear identity”. The ti account for the commuting flows of this integrable hierarchy. In this way, many interesting classical functions live under the same hat: characters of representations, Θ-functions of algebraic geometry, hypergeometric func- tions, certain integrals over classical Lie algebras or groups, Fredholm determinants, arising in statistical mechanics, in scattering and random matrix theory! They are all special instances of “τ-functions”. The point is that the probabilities or generating functions above, as functions of t1, t2, ... (after some minor renormalization) are precisely such τ-functions for the cor- responding integrable hierarchy and thus automatically satisfy a large set of equations. These probabilities are very special τ-functions: they happen to be a solution of yet another hierarchy of (linear) equations in the variables ti and the boundary points ci, J(2) J(2) namely k τ(t; c) = 0, where the k form -roughly speaking- a Virasoro-like algebra:

J(2), J(2) =(k ℓ) J(2) + ... k ℓ − k+ℓ h i The point is that each integrable hierarchy has a natural “vertex operator”, which auto- matically leads to a natural Virasoro algebra. Then, eliminating the partial derivatives in t from the two hierarchy of equations, the integrable and the Virasoro hierarchies, and finally setting t = 0, lead to PDE’s or ODE’s satisfied by the probabilities. In the table below, we give an overview of the different problems, discussed in this lecture, the relevant integrals in the second column and the different hierarchies satisfied by the integrals. To fix notation, , , refer to the Hermitian, symmetric and Hℓ Sℓ Tℓ symplectic ensembles, populated respectively by ℓ ℓ Hermitian matrices , symmetric × matrices and self-dual Hermitian matrices, with quaternionic entries. ℓ(E), ℓ(E), (E) are the corresponding set of matrices, with all spectral points belongingH S to E. Tℓ U(ℓ) and O(ℓ) are the unitary and orthogonal groups respectively. In the table below, i Vt(z) := V0(z)+ tiz , where V0(z) stands for the unperturbed problem; in the last integral V˜t(z) is a more complicated function of t1, t2, ... and z, to be specified later. P

4 1, p.5 §

Probability problem underlying t-perturbed corresponding integral, τ-function of integrable −→ hierarchies

∞ i P (M (E)) eTr( V (M)+ 1 tiM )dM Toda lattice n n(E) − ∈H H P R KP hierarchy

Tr( V (M)+ ∞ t M i) P (M (E)) e 1 i dM Pfaff lattice n n(E) − ∈ S S P R Pfaff-KP hierarchy

Tr( V (M)+ ∞ t M i) P (M (E)) e 1 i dM Pfaff lattice n n(E) − ∈T T P R Pfaff-KP hierarchy

P ((M1, M2) 2 (E) dM1dM2 2d-Toda lattice ∈ Hn Tr(Vt(M1) Vs(M2) cM1M2) KP-hierarchy (E ) (E )) eR− − − Hn 1 ×Hn 2

P (M (E)) det (I Kt(y, z)IEc (z)) KdV equation ∈H∞ − ( Fredholm determinant)

∞ i ¯ i Tr 1 (tiM siM ) longest increasing sequence U(ℓ) e − dM Toeplitz lattice P in random permutations R 2d-Toda lattice

Tr(xM+V˜t(M)) longest increasing sequence O(ℓ) e dM Toda lattice in random involutions R KP-hierarchy

Acknowledgment: These lectures represent joint work especially with (but also in- spired by) Mark Adler, Taka Shiota and Emil Horozov. Thanks also for many informa- tive discussions with Jinho Baik, Pavel Bleher, Edward Frenkel, Alberto Gr¨unbaum, Alexander Its, espacially Craig Tracy and Harold Widom, and with other participants in the semester at MSRI. I wish to thank Pavel Bleher, David Eisenbud and Alexander Its for organizing a truly stimulating and enjoyable semester at MSRI.

1 Matrix integrals, random matrices and permuta- tions

5 1, p.6 §

1.1 Tangent space to symmetric spaces and associated random matrix ensembles Random matrices provided a model for excitation spectra of heavy nuclei at high ex- citations (Wigner [76], Dyson [27] and Mehta [49]), based on the nuclear experimental data by Porter and Rosenzweig [56]; they observed that the occurrence of two levels, close to each other, is a rare event (level repulsion), showing that the spacing is not Poissonian, as one might expect from a naive point of view. Random matrix ideas play an increasingly prominent role in mathematics: not only have they come up in the spacings of the zeroes of the Riemann zeta function, but their relevance has been observed in the chaotic Sinai billiard and, more generally, in chaotic geodesic flows. Chaos seems to lead to the “spectral rigidity”, typical of the spectral distributions of random matrices, whereas the spectrum of an integrable system is random (Poisson)! (e.g., see Odlyzko [53] and Sarnak [60]). All these problems have led to three very natural random matrix ensembles: Hermi- tian, symmetric and symplectic ensembles. The purpose of this section is to show that these three examples appear very naturally as tangent spaces to symmetric spaces. A symmetric space G/K is given by a semi-simple Lie group G and a Lie group involution σ : G G such that → K = x G, σ(x)= x . { ∈ } Then the following identification holds: 1 G/K = gσ(g)− with g G , ∼ { ∈ } and the involution σ induces a map of the Lie algebra σ : g g such that (σ )2 =1, ∗ −→ ∗ with k = a g such that σ (a)= a g = k p with { ∈ ∗ } ⊕ p = a g such that σ (a)= a  { ∈ ∗ − } and [k, k] k, [k, p] p, [p, p] k. ⊂ ⊂ ⊂ 1 Then K acts on p by conjugation: kpk− p for all k K and p is the tangent space to G/K at the identity. The action of K on⊂p induces a∈ root space decomposition, with a being a maximal abelian subalgebra in p:

p = a + pα, with mα = dim pα. α ∆ X∈ Then, according to Helgason [35], the volume element on p is given by

dV = α(z)mα dz ...dz ,   1 n α ∆+ Y∈   6 1, p.7 §

where ∆+ is the set of positive roots; see [35, 36, 61, 62]. This will subsequently be worked out for the three so-called An-symmetric spaces. See also Sarnak’s MSRI-lecture [59] in these proceedings, who deals with more general symmetric spaces. I like to thank Chuu-Lian Terng for very helpful conversations on these matters.

Examples :

(i) Hermitian ensemble

1 1 Consider the non-compact symmetric space SL(n, C)/SU(n) with σ(g)=¯g⊤− . Then

SL(n, C)/SU(n) = gg¯⊤ g SL(n, C) { | ∈ } = positive definite matrices with det = 1 { } with

1 K = g SL(n, C) σ(g)= g = g SL(n, C) g− =g ¯⊤ = SU(n). { ∈ | } { ∈ | }

Then σ (a) = a¯⊤ and the tangent space to G/K is then given by the space p = n ∗ of Hermitian matrices− H

sl(n, C)= k p = su(n) , i.e., a = a + a , a su(n), a . ⊕ ⊕ Hn 1 2 1 ∈ 2 ∈ Hn If M , then the M , M and M (1 i < j n) are free variables, so that ∈ Hn ii ℜ ij ℑ ij ≤ ≤ Haar measure on M takes on the following form: ∈ Hn n dM := dM (d M d M ). (1.1.1) ii ℜ ij ℑ ij 1 1 i

A maximal abelian subalgebra a p = n is given by real diagonal matrices z = diag(z ,...,z ). Each M p = can⊂ be writtenH as 1 n ∈ Hn A A A M = e z e− , e K = SU(n), ∈ with2

A = (a (e e )+ ib (e + e )) k = su(n), a =0. (1.1.2) kℓ kℓ − ℓk kℓ kℓ ℓk ∈ ℓℓ 1 k ℓ n ≤X≤ ≤ Notice that e e and i(e + e ) k = su(n) and that kℓ − ℓk kℓ ℓk ∈ [e e , z]=(z z )(e + e ) p = kℓ − ℓk ℓ − k kℓ ℓk ∈ Hn 1The corresponding compact symmetric space is given by (SU(n) SU(n))/SU(n). 2e is the n n matrix with all zeroes, except for 1 at the (k,ℓ)th× entry. kℓ ×

7 1, p.8 §

[i(e + e ), z]=(z z )i(e e ) p = . (1.1.3) kℓ ℓk ℓ − k kℓ − ℓk ∈ Hn 3 Incidentally, this implies that ekℓ + eℓk and i(ekℓ + eℓk) are two-dimensional eigenspaces of (ad z)2 with eigenvalue (z z )2. From (1.1.2) and (1.1.3) it follows that ℓ − k [A, z]=(z z ) (a (e + e )+ ib (e e )) p = (1.1.4) ℓ − k kℓ kℓ ℓk kℓ kℓ − ℓk ∈ Hn 1 k<ℓ n ≤X≤ and thus, for small A, we have4

A A dM = d(e z e− ) = d(z + [A, z]+ ... ) n = dz d((z z )a )d((z z )b ), using (1.1.4) and (1.1.1) i ℓ − k kℓ ℓ − k kℓ 1 1 k<ℓ n ≤ ≤ Yn Y 2 = dzi∆n(z) dakℓdbkℓ. (1.1.5) 1 1 k<ℓ n Y ≤Y≤ Therefore ∆2(z) is also the Jacobian determinant of the map M (z, U), such that 1 → M = UzU − , and thus dM admits the decomposition in polar coordinates: ∈ Hn dM = ∆2 (z)dz ...dz dU, U SU(n). (1.1.6) n 1 n ∈ In random matrix theory, is endowed with the following probability, Hn trV (M) V (z) P (M dM)= c e− dM, ρ(dz)= e− dz, (1.1.7) ∈ n where dM is Haar measure (1.1.6) on and c is the normalizing factor. Since dM Hn n as in (1.1.6) contains dU and since the probability measure (1.1.7) only depends on the trace of V (M), dU completely integrates out. Given E R, define ⊂ (E) := M with all spectral points E R . (1.1.8) Hn { ∈ Hn ∈ ⊂ } ⊂ Hn Then 2 n Tr V (M) En ∆ (z) 1 ρ(dzk) − P (M n(E)) = cne dM = 2 n . (1.1.9) ∈ H n(E) Rn ∆ (z) 1 ρ(dzk) ZH R Q As the reader can find out from the excellent bookR by MehtaQ [49], it is well known that, if the probability P (M dM) satisfies the following two requirements: (i) in- ∈ 1 variance under conjugation by unitary transformations M UMU − , (ii) the random variables M , M , M , 1 i < j n are independent,7→ then V (z) is quadratic ii ℜ ij ℑ ij ≤ ≤ (Gaussian ensemble). 3ad x(y) := [x, y]. 4∆ (z)= (z z ) is the Vandermonde determinant. n i − j ≤ ≤ 1 i

(ii) Symmetric ensemble

5 1 Here we consider the non-compact symmetric space SL(n, R)/SO(n) with σ(g)= g⊤− . Then

SL(n, R)/SO(n) = gg⊤ g SL(n, R) { | ∈ } = positive definite matrices with det = 1 { } with

1 K = g SL(n, R) σ(g)= g = g SL(n, R) g⊤ = g− = SO(n). { ∈ | } { ∈ | }

Then σ (a)= a⊤ and the tangent space to G/K is then given by the space p = n of ∗ − S symmetric matrices, appearing in the decomposition of sl(n, R),

sl(n, R)= k p = so(n) , i.e., a = a + a , a so(n), a ⊕ ⊕ Sn 1 2 1 ∈ 2 ∈ Sn with Haar measure dM = dM on . ij Sn 1 i j n A maximal abelian subalgebra≤Y≤ ≤ a p = is given by real traceless diagonal matrices ⊂ Sn z = diag(z ,...,z ). Each M p = conjugates to a diagonal matrix z 1 n ∈ Sn A A A M = e z e− , e K = SO(n), A so(n). ∈ ∈ A calculation, analogous to example (i)(1.1.5) leads to

dM = ∆ (z) dz ...dz dU, U SO(n). n 1 n ∈ Random matrix theory deals with the following probability on : Sn trV (M) V (z) P (M dM)= c e− dM, ρ(dz)= e− dz, (1.1.10) ∈ n

with normalizing factor cn. Setting as in (1.1.8): n(E) n is the subset of matrices with spectrum E. Then S ⊂ S ∈ n Tr V (M) En ∆(z) 1 ρ(dzk) P (M n(E)) = cne− dM = | | n . (1.1.11) ∈ S n(E) Rn ∆(z) 1 ρ(dzk) ZS R | | Q As in the Hermitian case, P (M dM) is Gaussian,R if P (M QdM) satisfies ∈ ∈ 1 (i) invariance under conjugation by orthogonal conjugation M OMO− , (ii) Mii, Mij (i < j) are independent random variables. → 5The compact version is given by SU(n)/SO(n).

9 1, p.10 §

(iii) Symplectic ensemble

6 1 1 Consider the non-compact symmetric space SU ∗(2n)/USp(n) with σ(g)= Jg⊤− J − , where J is the 2n 2n matrix: × 0 1 1 0  −  0 1    1 0  2 J :=   with J = I, (1.1.12)  −  −  0 1     1 0   −   .   ..      and

1 G = SU ∗(2n)= g SL(2n, C) g = JgJ¯ − , { ∈ | }

K = g SU ∗(2n) σ(g)= g := Sp(n, C) U(2n) { ∈ | } ∩ 1 = g SL(2n, C) g⊤Jg = J g SL(2n, C) g− =g ¯⊤ { ∈ }∩{ ∈ } 1 1 = g SL(2n, C) g− =g ¯⊤and g = JgJ¯ − { ∈ } =: USp(n).

1 Then, σ (a)= Ja⊤J − and ∗ −

k = a su∗(2n) σ (a)= a = sp(n, C) u(2n) { ∈ ∗ } ∩ 2n 2n 1 = a C × a⊤ = a,¯ a = JaJ¯ − { ∈ − }

p = a su∗(2n) σ (a)= a = su∗(2n) iu(2n) { ∈ ∗ − } ∩ 2n 2n 1 = a C × a⊤ =¯a, a = JaJ¯ − { ∈ } (0) (1) Mkℓ Mkℓ ¯ C2 2 = M =(Mkℓ)1 k,ℓ n, Mkℓ = with Mℓk = Mkℓ⊤ ×  ≤ ≤  (1) (0)  ∈   M¯ M¯  − kℓ kℓ = self-dual n n Hermitean matrices, with quaternionic entries ∼ { × }  =: . T2n

The condition on the 2 2 matrices Mkℓ implies that Mkk = MkI, with Mk R and the 2 2 identity I. Notice× USp(n) acts naturally by conjugation on the tangent∈ space × 6The corresponding compact symmetric space is SU(2n)/Sp(n).

10 1, p.11 §

p to G/K. Haar measure on is given by T2n n (0) ¯ (0) (1) ¯ (1) dM = dMk dMkℓ dMkℓ dMkℓ dMkℓ , (1.1.13) 1 1 k<ℓ n Y ≤Y≤ since these M are the only free variables in the matrix M . A maximal abelian ij ∈ T2n subalgebra in p is given by real diagonal matrices of the form z = diag(z1, z1, z2, z2, ..., zn, zn). Each M p = can be written as ∈ T2n A A A M = e ze− , e K = USp(n), (1.1.14) ∈ with (a , b ,c ,d R) kℓ kℓ kℓ kℓ ∈ A = a (e(0) e(0))+ b (e(1) + e(1))+ c (e(2) e(2))+ d (e(3) + e(3)) k kℓ kℓ − ℓk kℓ kℓ ℓk kℓ kℓ − ℓk kℓ kℓ ℓk ∈ 1 k ℓ n (1.1.15) ≤X≤ ≤ in terms of the four 2 2 matrices 7 × 1 0 i 0 0 1 0 i e(0) = , e(1) = , e(2) = , e(3) = . 0 1 0 i 1 0 i 0    −   −    Since e(0) e(0), z = (z z )(e(0) + e(0)) p kℓ − ℓk ℓ − k kℓ ℓk ∈ (1) (1) (1) (1) ekℓ + eℓk , z = (zℓ zk)(ekℓ eℓk ) p − − ∈ (2) (2) (2) (2) ekℓ eℓk , z = (zℓ zk)(ekℓ + eℓk ) p − − ∈ (3) (3) (3) (3) ekℓ + eℓk , z = (zℓ zk)(ekℓ eℓk ) p, (1.1.16) − − ∈ [A, z] p has the following form: it has 2 2 zero blocks along the diagonal and from (1.1.16)∈ and (1.1.15), ×

akℓ + ibkℓ ckℓ + idkℓ ((k,ℓ)th block in [A, z]) = (zℓ zk) , (k<ℓ). (1.1.17) − ckℓ + idkℓ akℓ ibkℓ  − −  Therefore, using (1.1.17), Haar measure dM on equals T2n A A dM = d(e ze− ) = d(I + A + ... )z(I A + ... ) − = d(z + [A, z]+ ... ) = dz d((z z )(a + ib ))d((z z )(a ib )) k ℓ − k kℓ kℓ ℓ − k kℓ − kℓ 1 k n 1 k<ℓ n ≤Y≤ ≤Y≤ d((z z )(c + id ))d((z z )( c + id )) ℓ − k kℓ kℓ ℓ − k − kℓ kℓ = ∆4(z)dz dz 4da db dc dd . 1 ··· n kℓ kℓ kℓ kℓ 1 k<ℓ n ≤Y≤ 7e(i) in (1.1.15) refers to putting the 2 2 matrix e(i) at place (k,ℓ). kℓ × 11 1, p.12 §

As before, define 2n(E) 2n as the subset of matrices with spectrum E and define the probability:T ⊂ T ∈

4 n Tr V (M) En ∆ (z) 1 ρ(dzk) − P (M 2n(E)) = cne dM = 4 n . (1.1.18) ∈ T 2n(E) Rn ∆ (z) 1 ρ(dzk) ZT R Q R Q Remark: Notice 2n is called the symplectic ensemble, although the matrices in p = 2n are not at all symplectic;T but rather the matrices in k are. T

1.2 Infinite Hermitian matrix ensembles Consider now he limit of the probability

∆2(z) n ρ(dz ) P (M (E)) = En 1 k , when n . (1.2.1) ∈ Hn ∆2(z) n ρ(dz ) ր∞ RRn Q1 k Dyson [27] (see also Mehta [49]) usedR the followingQ trick, to circumvent the problem of dealing with -fold integrals. Using the orthogonality of the monic orthogonal ∞ R 2 2 polynomials pk = pk(z) for the weight ρ(dz) on , and the L -norms hk = R pk(z)ρ(dz) of the p ’s, one finds, using (det A)2 = det(AA ), k ⊤ R n 2 ∆ (z) ρ(dzi) Rn 1 Z Y n

= det(pi 1(zj))1 i,j n det(pk 1(zℓ))1 k,ℓ n ρ(dzk) Rn − ≤ ≤ − ≤ ≤ Z k=1 n Y π+π′ = ( 1) pπ(k) 1(zk)pπ′(k) 1(zk)ρ(dzk) − − ′ − R π,π σn k=1 X∈ Y Z n 1 n 1 − − 2 = n! pk(z)ρ(dz)= n! hk. (1.2.2) R 0 0 Y Z Y For the integral over an arbitrary subset E R, one stops at the second equality, since ⊂ the pn’s are not necessarily orthogonal over E. This leads to the probability (1.2.1),

P (M (E))) ∈ Hn 1 n = n det pj 1(zk)pj 1(zℓ) ρ(dzi) − − n! 1 hi 1 En ! − Z 1 j n 1 k,ℓ n 1 ≤X≤ ≤ ≤ Y 1 Q n = det(Kn(zk, zℓ))1 k,ℓ n ρ(dzi), (1.2.3) n! n ≤ ≤ E 1 Z Y 12 1, p.13 §

in terms of the kernel n pj 1(y) pj 1(z) Kn(y, z) := − − . (1.2.4) j=1 hj 1 hj 1 X − − p p The orthonormality relations of the pk(y)/√hk lead to the reproducing property for the kernel Kn(y, z):

Kn(y, z)Kn(z,u)ρ(dz)= Kn(y,u), Kn(z, z)ρ(dz)= n. (1.2.5) R R Z Z n k n k Upon replacing E by dz R − in (1.2.3), upon integrating out all the remaining 1 i × variables zk+1, ..., zn and using the reproducing property (1.2.5), one finds the n-point correlation function Q

P (one eigenvalue in each [zi, zi + dzi], i =1, ..., k) k

= cn det (Kn(zi, zj))1 i,j k ρ(dzi). (1.2.6) ≤ ≤ 1 Y Finally, by Poincar´e’s formula for the probability P E , the probability that no ∪ i spectral point of M belongs to E is given by a Fredholm determinant
P (M (Ec)) = det(I λKE) ∈ Hn − n k ∞ k E = 1+ ( λ) det Kn (zi, zj) ρ(dzi), − 1 i,j k z1 ... zk ≤ ≤ 1 Xk=1 Z ≤ ≤   Y E for the kernel Kn (y, z)= Kn(y, z)IE(z). Wigner’s semi-circle law: For this ensemble (defined by a large class of ρ’s, in partic- •ular for the Gaussian ensemble) and for very large n, the density of eigenvalues tends to Wigner’s semi-circle distribution on the interval [ √2n, √2n]: − 1 = √2n z2dz, z √2n π − | | ≤ density of eigenvalues   =0, z > √2n. | | Bulk scaling limit: From the formula above, it follows that the average number of eigenvalues• per unit length near z = 0 (“the bulk”) is given by √2n/π and thus the average distance between two consecutive eigenvalues is given by π/√2n. Upon using this rescaling, one shows ([43, 48, 52, 55, 39])

π πx πy sin π(x y) lim Kn , = − (Sine kernel) n √2n √2n √2n π(x y) ր∞   − 13 1, p.14 §

and

( 1)k ∂ k P (exactly k eigenvalues [0, a]) = − det(I λKI[0,a]) ∈ k! ∂λ −   λ=1

with πa f(x; λ) det(I λKI ) = exp dx, (1.2.7) − [0,a] x Z0 where f(x, λ) is a solution to the following differential equation, due to the pioneering work of Jimbo, Miwa, Mori, Sato [39], (′ = ∂/∂x)

2 2 λ (xf ′′) = 4(xf ′ f)( f ′ xf ′ + f) , with f(x; λ) = x for x 0. − − − ∼ −π ≃ (Painlev´eV) (1.2.8)

Edge scaling limit: Near the edge √2n of the Wigner semi-circle, the scaling is √2n1/6 • and thus the scaling is more subtle: (see [21, 30, 51, 49, 64]) u y = √2n + , (1.2.9) √2n1/6 and so for the kernel Kn as in (1.2.4), with the pn’s being Hermite polynomials, 1 u v lim Kn √2n + , √2n + = K(u, v), n √2n1/6 √2n1/6 √2n1/6 ր∞   where ∞ ∞ iux x3/3 K(u, v)= A(x + u)A(x + v)dx, A(u)= e − dx. 0 Z Z−∞ Relating y and u by (1.2.9), the statistics of the largest eigenvalue for very large n is governed by the function,

2 λmax P (λmax y) = P 2n 3 1 u , for n , ≤ √2n − ≤ ր∞     ∞ 2 = det(I KI( ,u]) = exp (α u)g (α)dα , − −∞ − −  Zu  with g(x) a solution of

3 g′′ = xg +2g 3 2 2 (Painlev´eII) (1.2.10) e− 3 x ( g(x) = 1/4 for x . ∼ − 2√πx ր∞ The latter is essentially the asymptotics of the Airy function. In section 5, I shall derive, via Virasoro constraints, not only this result, due to Tracy-Widom [64], but

14 1, p.15 §

also a PDE for the probability that the eigenvalues belong to several intervals, due to Adler-Shiota-van Moerbeke [11, 12]. Hard edge scaling limit: Consider the ensemble of n n random matrices for the La- • × ν/2 z/2 guerre probability distribution, thus corresponding to (1.1.9) with ρ(dz)= z e− dz. One shows the density of eigenvalues near z = 0 is given by 4n for very large n. At this edge, one computes for the kernel (1.2.4) with Laguerre polynomials pn [52, 30]:

1 (ν) u v (ν) lim Kn , = K (u, v), (1.2.11) n 4n 4n 4n ր∞   (ν) where K (u, v) is the Bessel kernel, with Bessel functions Jν,

1 1 K(ν)(u, v) = xJ (xu)J (xv)dx 2 ν ν Z0 J (u)√uJ ′ (v) J (√v)√vJ ′ (√u) = ν ν − ν ν . (1.2.12) 2(u v) − Then x f(u) P ( no eigenvalues [0, x]) = exp du , ∈ − u  Z0  with f satisfying

2 2 2 (xf ′′) 4 xf ′ f f ′ + (x ν )f ′ f f ′ =0. (Painlev´eV) (1.2.13) − − − − This result due to Tracy-Widom
[65] and a more
general statement, due to [11, 12] will be shown using Virasoro constraints in section 5.

1.3 Integrals over classical groups The integration on a compact semi-simple simply connected Lie group G is given by the formula (see Helgason [36])

1 α(iH) 1 H f(M)dM = 2 sin dt f(utu− )du, t = e , (1.3.1) G W T 2 U Z | | Z α ∆ Z Y∈ where A G is a maximal subgroup, with g and a being the Lie algebras of G and A. Let du and⊂ dt be Haar measures on G, A respectively such that

dt = du = 1; ZA ZU ∆ denotes the set of roots of g with respect to a; W is the order of the Weyl group of G. | |

15 1, p.16 §

∞ t Tr M i Integration formula (1.3.1) will be applied to integrals of f = e 1 i over the groups SO(2n), SO(2n+1) and Sp(n). Their Lie algebras (over C) areP given respectively by dn, bn, cn, with sets of roots: (e.g., see [20]) ∆ = εe , 1 i k, (e + e ), (e e ), 1 i < j n , n {± i ≤ ≤ ± i j ± i − j ≤ ≤ } with ε = 0 for dn = so(2n) ε = 1 for bn = so(2n + 1) ε = 2 for cn = sp(n).

Setting H = iθ, we have, in view of formula (1.3.1), α(iH) 2 sin dt 2 α ∆ Y∈ 2 θ θ θ +θ n j − k j k cn 1 j

n n ∞ t Tr M k ∞ t n P(eikθj +e−ikθj ) 2 ∞ t cos kθ 2 t T (z ) e 1 k = e 1 k j=1 = e k=1 k j = e k k j , (1.3.2) P P P j=1 P j=1 P Y Y

16 1, p.17 §

where Tn(z) are the Tchebychev polynomials, defined by Tn(cos θ) := cos nθ; in partic- ular T1(z)= z. iθ iθ For M SO(2n + 1), the eigenvalues are given by 1, e j and e− j , 1 j n, ∈ ≤ ≤ which is responsible for the extra-exponential e ti appearing in (1.3.2). Before listing various integrals, define the JacobiP weight ρ (z)dz := (1 z)α(1 + z)βdz, (1.3.3) αβ − and the formal sum ∞ g(z) := 2 tiTi(z). 1 X The arguments above lead to the following integrals, originally due to H. Weyl [75], and in its present form, due to Johansson [40]; besides the integrals over SO(k) = O+(k), the integrals over O (k) and U(n) will also be of interest in the theory of random − permutations:

n ∞ i 1 titrM 2 g(zk) e dM = ∆n(z) e ρ( 1 , 1 )(zk)dzk n 2 2 O(2n)+ P [ 1,1] − − Z Z − k=1 Y n ∞ i ∞ 1 titrM 1 ti 2 g(zk) e dM = e ∆n(z) e ρ( 1 , 1 )(zk)dzk n 2 2 O(2n+1)+ P P [ 1,1] − Z Z − k=1 n Y ∞ i titrM 2 g(zk) e 1 dM = ∆n(z) e ρ 1 1 (zk)dzk. ( 2 , 2 ) Sp(n) P [ 1,1]n Z Z − k=1 Y n 1 ∞ i ∞ − 1 titrM 1 2t2i 2 g(zk) e dM = e ∆n 1(z) e ρ( 1 , 1 )(zk)dzk n−1 − 2 2 O(2n)− P P [ 1,1] Z Z − k=1 nY ∞ i ∞ i 1 titrM 1 ( 1) ti 2 g(zk) e dM = e − ∆n(z) e ρ( 1 , 1 )(zk)dzk n 2 2 O(2n+1)− P P [ 1,1] − Z Z − Yk=1

n ∞ i i ∞ i −i tr(tiM siM¯ ) 1 2 (tiz siz ) dzk e 1 − dM = ∆n(z) e 1 k− k U(n) P n! (S1)n | | P 2πizk Z Z Yk=1 (1.3.4)

1.4 Permutations and integrals over groups

0 Let Sn be the group of permutations π and S2n the subset of fixed-point free involutions 0 0 2 0 π (i.e., (π ) = I and π (k) = k for 1 k 2n ). Put the uniform distribution on Sn 0 6 ≤ ≤ and S2n ; i.e., all permutations or involutions have equal probability: 2nn! P (π )=1/n! and P (π0 )= ; (1.4.1) n 2n (2n)!

17 1, p.18 §

0 0 πn refers to a permutation in Sn and π2n to an involution in S2n. An increasing subsequence of π S or S0 is a sequence 1 j < ... < j n, such ∈ n n ≤ 1 k ≤ that π(j1) < ... < π(jk). Define

L(πn) = length of the longest increasing subsequence of πn . (1.4.2)

Example: for π = (3, 1, 4, 2, 6, 7, 5), we have L(π7) = 4. Around 1960 and based on Monte-Carlo methods, Ulam [70] conjectured that

E(L ) lim n = c exists. n →∞ √n

An argument of Erd¨os & Szekeres [28], dating back from 1935 showed that E(Ln) 1 ≥ 2 √n 1, and thus c 1/2. In ’72, Hammersley [33] showed rigorously that the limit exists.− Logan and Shepp≥ [46] showed the limit c 2, and finally Vershik and Kerov ≥ [74] that c = 2. In 1990, I. Gessel [31] showed that the following generating function is the determinant of a Toeplitz matrix:

∞ n 2π t 2√t cos θ i(k m)θ P (L ℓ) = det e e − dθ . (1.4.3) n! n ≤ n=0 0 0 k,m ℓ 1 X Z  ≤ ≤ − The next major contribution was due to Johansson [41] and Baik-Deift-Johansson [17], who prove that for arbitrary x R, we have a ”law of large numbers” and a ”central limit theorem”, where F (x) is the∈ statistics (1.2.10),

L L 2√n lim n =1, and P n − x F (x), for n . n 2√n n1/6 ≤ −→ −→ ∞ →∞   A next set of ideas is due to Diaconis & Shashahani [26], Rains [57, 58], Baik & Rains [18]. For a nice state-of-the-art account, see Aldous & Diaconis [14]. An illustration is contained in the following proposition; the first statement is essentially Gessel’s and the next statement is due to [26, 58, 18].

Proposition 1.1 The following holds

∞ tn P (L(π ) ℓ) = e√t Tr(M+M¯ )dM (1.4.4) n! n ≤ n=0 U(ℓ) X Z dθk = eiθj eiθk 2 e2√t cos θk . ℓ | − | 2π [0,2π] 1 j

∞ (t2/2)n P (L(π0 ) ℓ)= et Tr M dM. (1.4.5) n! 2n ≤ n=0 O(ℓ) X Z 18 1, p.19 §

The proof of this statement will be sketched later. The connection with integrable systems goes via the following chain of ideas:

Combinatorics

↓ Robinson-Schensted-Knuth correspondence

↓ Theory of symmetric polynomials

↓ Integrals over classical groups

↓ Integrable systems All the arrows, but the last one, will be explained in this section; the last arrow will be discussed in sections 7 and 8. We briefly sketch a few of the basic well known facts going into these arguments. They can be found in MacDonald [47], Knuth [45], Aldous- Diaconis [14]. Useful facts on symmetric functions, applicable to integrable theory, can be found in the appendix to [1]. Let me mention a few of these facts:

A Young diagram λ is a finite sequence of non-increasing, non-negative integers • λ λ ... λ 0; also called a partition of n = λ := λ + ... + λ , with 1 ≥ 2 ≥ ≥ ℓ ≥ | | 1 ℓ λ being the weight. It can be represented by a diagram, having λ boxes in the | | 1 first row, λ2 boxes in the second row, etc..., all aligned to the left. A dual Young ˆ ˆ ˆ diagram λ = (λ1 λ2 ...) is the diagram obtained by flipping the diagram λ about its diagonal.≥ ≥

A Young tableau of shape λ is an array of positive integers a (at place (i, j) in • ij the Young diagram) placed in the Young diagram λ, which are non-decreasing from left to right and strictly increasing from top to bottom.

A standard Young tableau of shape λ is an array of integers 1, ..., n placed in the • Young diagram, which are strictly increasing from left to right and from top to bottom. The number of Young tableaux of a given shape λ =(λ ... λ ) is 1 ≥ ≥ m

19 1, p.20 §

8 given by a number of formulae (for the Schur polynomial sλ, see below)

f λ = # standard tableaux of shape λ { } = coefficient of x1x2 ...xn in sλ(x) λ ! 1 = | | = λ ! det (1.4.6) hλ | | (λ i + j)! all i,j ij  i −  m 1 = Qλ ! (h h ) , with h := λ i + m, m := λˆ . | | i − j h ! i i − 1 1 i

sλ(x1, x2, ...) := xaij . (1.4.7) a tableaux of λ ij { ij } X Y

The linear space Λn of symmetric polynomials in x1, ..., xn with rational coefficients • comes equipped with the inner product

n 1 2 dzk f,g = f(z1, ..., zn)g(¯z1, ..., z¯n) zk zℓ h i n! n | − | 2πizk (S1) 1 k<ℓ n 1 Z ≤Y≤ Y = f(M)g(M¯ )dM. (1.4.8) ZU(n) An orthonormal basis of the space Λ is given by the Schur polynomials s (x , ..., x ), • n λ 1 n in which the numbers aij are restricted to 1, ..., n. Therefore, each symmetric func- tion admits a “Fourier series”

f(x , ..., x )= f,s s (x , ..., x ), with s ,s ′ = δ ′ . (1.4.9) 1 n h λi λ 1 n h λ λ i λλ λ Xwith λˆ1 n ≤ In particular, one proves (see (1.4.6) for the definition of f λ)

k λ (x1 + ... + xn) = f sλ, (1.4.10) |λ|=k ˆ λX1≤n

If λ =(λ ... λ > 0), with9 λˆ = ℓ > n, then obviously s = 0. 1 ≥ ≥ ℓ 1 λ 8 λ ˆ hij := λi + λj i j + 1 is the hook length of the i, jth box in the Young diagram; i.e., the length of the hook formed− by− drawing a horizontal line emanating from the center of the box to the right and a vertical line emanating from the center of the box to the bottom of the diagram. 9 Remember, from the definition of the dual Young diagram, that λˆ1 = the length of the first column of λ

20 1, p.21 §

Robinson-Schensted-Knuth correspondence: There is a 1-1 correspondence • (P, Q), two standard Young S tableaux from 1,... ,n, where n −→    P and Q have the same shape    Given a permutation i1, ..., in, the correspondence constructs two standard Young tableaux P, Q having the same shape λ. This construction is inductive. Namely, having obtained two equally shaped Young diagrams Pk, Qk from i1, ..., ik, with the numbers (i1, ..., ik) in the boxes of Pk and the numbers (1, ..., k) in the boxes of Qk, one creates a new diagram Qk+1, by putting the next number ik+1 in the first row of P , according to the following rule:

(i) if i all numbers appearing in the first row of P , then one creates a new k+1 ≥ k box with ik+1 in that box to the right of the first column,

(ii) if not, place ik+1 in the box (of the first row) with the smallest number higher than ik+1. That number then gets pushed down to the second row of Pk according to the rule (i) or (ii), as if the first row had been removed. The diagram Q is a bookkeeping device; namely, add a box (with the number k + 1 in it) to Qk exactly at the place, where the new box has been added to Pk. This produces a new diagram Qk+1 of same shape as Pk+1. The inverse of this map is constructed essentially by reversing the steps above.

Example: π = (5, 1, 4, 3, 2) S , ∈ 5 5 1 14 13 12 55 4 3 5 4 5

1 1 13 13 13 22 2 2 4 4 5 1 2 1 3 3 2 Hence π (P (π), Q(π)) =   ,   −→ 4 4  5   5      and so L5(π) = 2 = #columns ofPor Q.   

The Robinson-Schensted-Knuth correspondence has the following properties

1 π (P, Q), then π− (Q, P ) • 7→ 7→ 21 1, p.22 §

length (longest increasing subsequence of π) = # (columns in P ) • length (longest decreasing subsequence of π) = # (rows in P ) • π2 = I, then π (P,P ) • 7→ π2 = I, with k fixed points, then P has exactly k columns of odd length. • (1.4.11)

From representation theory (see Weyl [75] and especially Rains [57]), one proves:

Lemma 1.2 The following perpendicularity relations hold: (i) s (M)s (M¯ )dM = s ,s = δ λ µ h λ µi λµ ZU(n)

(ii) s (M)dM = 1 for λ =(λ ... λ 0), k n, λ even λ 1 ≥ ≥ k ≥ ≤ i ZO(n) = 0 otherwise.

(iii) s (M)dM = 1 for λˆ even, λˆ 2n , λ i 1 ≤ ZSp(n) = 0 otherwise. (1.4.12)

Proof of Proposition 1.1 : On the one hand, (x + ... + x )k, (x + ... + x )k h 1 n 1 n i = f λf µ s ,s h λ µi |λ|=|µ|=k ˆ λ1X,µˆ1≤n = (f λ)2 |λ|=k ˆ λX1≤n

= (f λ)2 |λ|=k λX1≤n = # (P, Q), standard Young tableaux, each of arbitrary shape λ { with λ = k, λ n | | 1 ≤ } = # π S such that L(π ) n . (1.4.13) { k ∈ k k ≤ }

On the other hand, notice that, upon setting θ = θ′ + θ for 2 j n, the expres- j j 1 ≤ ≤ sion eiθj eiθk 2 is independent of θ . Then, setting z = eiθk , one computes: | − | 1 k 1 j

22 1, p.23 §

(x + ... + x )k, (x + ... + x )ℓ h 1 n 1 n i 1 k ℓ iθj iθk 2 = (z1 + ... + zn) (¯z1 + ... +¯zn) e e dθ1 ...dθn n! n | − | [0,2π] 1 j

(tr(M + M¯ ))kdM ZU(n) k j k j = ((trM) (trM) − )dM j 0 j k   U(n) ≤X≤ Z 0, if k is odd (because then j = k j for all 0 j k) 6 − ≤ ≤ = k (1.4.15)  trM kdM, if k is even. k/2 U(n) | |    R Combining the three identities (1.4.13), (1.4.14) and (1.4.15) leads to

1 2k − # π S such that L(π ) n = (Tr(M + M¯ ))2kdM. (1.4.16) { k ∈ k k ≤ } k   ZU(n) Finally

23 1, p.24 §

∞ tn P (L(π ) ℓ) n! n ≤ n=0 X n ∞ t # πn Sn L(πn) ℓ = { ∈ ≤ } n! n! n=0 X 1 ∞ tn 2n − = (tr(M + M¯ ))2ndM (n!)2 n n=0 U(ℓ) X   Z ∞ (√t)2n = (tr(M + M¯ ))2ndM (2n)! n=0 U(ℓ) X Z = e√t Tr(M+M¯ )dM ZU(ℓ) 1 √t(z +z−1+...+z +z−1) iθ iθ 2 dθk = e 1 1 ℓ ℓ e j e k , ℓ! ℓ | − | 2π [0,2π] 1 j

1 ℓ dz √t(zk+¯zk) k = ∆ℓ(z)∆ℓ(¯z) e ℓ! (S1)ℓ 2πizk Z Yk=1   1 ℓ dz k 1 m 1 √t(zk+¯zk) k = det zσ(−m)z¯σ(−m) e ℓ! (S1)ℓ 1 k,m ℓ 2πizk σ Sℓ ≤ ≤ k=1   Z X∈   Y 1 dz k 1 m 1 √t(zk+¯zk) k = det zk− z¯k − e ℓ! 1 2πizk σ S S 1 k,m ℓ ∈ ℓ Z  ≤ ≤ X 2π 2√t cos θ i(k m)θ = det e e − dθ , 0 1 k,m ℓ Z  ≤ ≤

confirming Gessel’s result (1.4.3). The proof of the second relation (1.4.5) of Proposition 1.1 is based on the following computation:

24 1, p.25 §

k λ (Tr M) dM = f sλ(M)dM, using (1.4.10), O(n) |λ|=k O(n) Z ˆ Z λX1≤n

= f λ , using Lemma 1.2, λ =k | | λˆ1 n λi Peven≤

= f λ , using duality, λ =k |λ1| n ≤ λˆi Peven (P,P ),P standard Young tableau of shape λ = # with λ = k, λ n, λˆ even  | | 1 ≤ i  = # π0 S0, no fixed points and L(π0) n . k ∈ k k ≤  (1.4.17)

In the last equality, we have used property (1.4.11): an involution has no fixed points iff all columns of P have even length. Since all columns λˆi have even length, it follows that λ = k is even and then only is (Tr M)kdM > 0; otherwise = 0 . Finally, one | | O(n) computes, R ∞ (t2/2)k P L(π0 ) n, π0 S0 k! 2k ≤ 2k ∈ 2k k=0 X
∞ t2k 2kk! = # π0 S0 , L(π0 ) n , using (1.4.1), 2kk! (2k)! { 2k ∈ 2k 2k ≤ } Xk=0 ∞ tk = # π0 S0, L(π0) n k! { k ∈ k k ≤ } Xk=0 ∞ tk = (Tr M)kdM , using (1.4.17), k! O(n) Xk=0 Z = et Tr M dM, ZO(n) ending the proof of Proposition 1.1.

25 2, p.26 § 2 Integrals, vertex operators and Virasoro relations

In section 1, we discussed random matrix problems over different finite and infinite ma- trix ensembles, generating functions for the statistics of the length of longest increasing sequences in random permutations and involutions. One can also consider two Hermi- tian random matrix ensembles, coupled together. All those problems lead to matrix integrals or Fredholm determinants, which we list here: (β =2, 1, 4)

n T rV (M) β e− dM = cn ∆n(z) ρ(zk)dzk • n | | n(E), n(E) or n(E) E 1 ZH S T Z Y

1 2 2 Tr(M +M 2cM1M2) dM1dM2 e− 2 1 2 − 2 • (E1 E2) Z ZHn × ex Tr M dM • ZO(n) e√x Tr(M+M¯ )dM • ZU(n) det I λK(y, z)I (z) with K(y, z) as in (1.2.4). • − E (2.0.1)
The point is that each of these quantities admit a natural deformation, by inserting time variables t1, t2, ... and possibly a second set s1,s2, ..., in a seemingly ad hoc way. Each of these integrals or Fredholm determinant is then a fixed point for a natural vertex operator, which generates a Virasoro-like algebra. These new integrals in t1, t2, ... are all annihilated by the precise subalgebra of the Virasoro generators, which annihilates τ0. This will be the topic of this section.

2.1 Beta-integrals 2.1.1 Virasoro constraints for Beta-integrals

V (z) Consider weights of the form ρ(z)dz := e− dz on an interval F = [A, B] R, with ⊆ rational logarithmic derivative and subjected to the following boundary conditions:

i ρ′ g 0∞ biz k = V ′ = = , lim f(z)ρ(z)z = 0 for all k 0, (2.1.1) − ρ f ∞ a zi z A,B ≥ P0 i → and a disjoint union of intervals,P

2r

E = [c2i 1,c2i] F R. (2.1.2) − ⊂ ⊂ 1 [

26 2, p.27 §

These data define an algebra of differential operators 2r ∂ = ck+1f(c ) . (2.1.3) Bk i i ∂c 1 i X Take the first type of integrals in the list (2.0.1) for general β > 0, thus general- izing the integrals, appearing in the probabilities (1.1.9),(1.1.11) and (1.1.18). Con- sider t-deformations of such integrals, for general (fixed) β > 0: (t := (t1, t2, ...), c =(c1,c2, ..., c2r) and z =(z1, ..., zn)) n ∞ i β tiz In(t, c; β) := ∆n(z) e 1 k ρ(zk)dzk for n> 0. (2.1.4) En | | P Z kY=1   The main statement of this section is Theorem 2.1, whose proof will be outlined in the next subsection. The central charge (2.1.9) has already appeared in the work of Awata et al. [16]. Theorem 2.1 (Adler-van Moerbeke [3, 6]) The multiple integrals n ∞ i β tiz In(t, c; β) := ∆n(z) e 1 k ρ(zk)dzk , for n> 0 (2.1.5) En | | P Z kY=1   and n ∞ i 4 4/β tiz In(t, c; ) := ∆n(z) e 1 k ρ(zk)dzk , for n> 0, (2.1.6) β En | | P Z kY=1   with I =1, satisfy respectively the following Virasoro constraints10 for all k 1: 0 ≥ − + a βJ(2) (t, n) b βJ(1) (t, n) I (t, c; β)=0, −Bk i k+i,n − i k+i+1,n n i 0 ! X≥   βt 2n βb βt 2n 4 + a βJ(2) , + i βJ(1) , I (t, c; )=0, −Bk i k+i,n − 2 − β 2 k+i+1,n − 2 − β n β i 0  ! X≥

(2.1.7)

in terms of the coefficients ai, bi of the rational function ( log ρ)′ and the end points ci Z − βJ(2) βJ(1) of the subset E, as in (2.1.1) to (2.1.2). For all n , the k,n(t, n) and k,n(t, n) form a Virasoro and a Heisenberg algebra respectively,∈ interacting as follows k3 k βJ(2) βJ(2) βJ(2) k,n, ℓ,n = (k ℓ) k+ℓ,n + c − δk, ℓ − 12 − h i   βJ(2) βJ(1) βJ(1) k,n, ℓ,n = ℓ k+ℓ,n + c′k(k + 1)δk, ℓ. − − h i k βJ(1) βJ(1) k,n, ℓ,n = δk, ℓ, (2.1.8) β − 10 When E equalsh the whole rangei F , then the ’s are absent in the formulae (2.1.7). Bk 27 2, p.28 §

with central charge

1/2 1/2 2 β β − 1 1 c =1 6 and c′ = . (2.1.9) − 2 − 2 β − 2     !  

βJ(2) Remark 1: The k,n’s are defined as follows: β β βJ(2) = : βJ(1) βJ(1) :+ 1 (k + 1) βJ(1) kJ(0) . (2.1.10) k,n 2 i,n j,n − 2 k,n − k,n i+Xj=k     Componentwise, we have βJ(1) β (1) (0) βJ(0) (0) k,n(t, n)= Jk + nJk and k,n = nJk = nδ0k and βJ(2) k,n(t, n) β β β = βJ (2) + nβ +(k + 1)(1 ) βJ (1) + n (n 1) +1 J (0), 2 k − 2 k − 2 k       where

β (1) ∂ 1 Jk = + ( k)t k (2.1.11) ∂tk β − − 2 β (2) ∂ 2 ∂ 1 Jk = + iti + 2 itijtj. ∂ti∂tj β ∂tj β i+j=k i+j=k i j=k X −X −X− βJ(2) We put n explicitly in ℓ,n(t, n) to indicate the nth component contains n explicitly, besides t. Note that for β = 2, (2.1.10) becomes particularly simple: βJ(2) 2J(1) 2J(1) k,n = : i,n j,n : . β=2 i+j=k X Remark 2: The Heisenberg and Virasoro generators satisfy the following duality prop- erties:

4 (2) (2) βt 2n β J t, n = βJ , , n Z ℓ,n ℓ,n − 2 − β ∈     4 (1) β β (1) βt 2n β J t, n = J , , n> 0. (2.1.12) ℓ,n − 2 ℓ,n − 2 − β     βJ(2) In (2.1.7), ℓ,n ( βt/2, 2n/β) means that the variable n, which appears in the nth component, gets− replaced− by 2n/β and t by βt/2. − Remark 3: Theorem 2.1 states that the integrals (2.1.5) and (2.1.6) satisfy two sets of differential equations (2.1.7) respectively. Of course, the second integral also satisfies the first set of equations, with β replaced by 4/β.

28 2, p.29 §

2.1.2 Proof: β-integrals as fixed points of vertex operators Theorem 2.1 can be established by using the invariance of the integral under the trans- k+1 formation zi zi +εf(zi)zi of the integration variables. However, the most transpar- ent way to prove7→ Theorem 2.1 is via vector vertex operators, for which the β-integrals are fixed points. This is a technique which has been used by us, already in [2]. In- deed, define the (vector) vertex operator, for t = (t , t , ...) C∞, u C, and setting 1 2 ∈ ∈ χ(z) := (1,z,z2, ...).

−i ∞ i ∞ u ∂ 1 tiu β 1 i ∂t β Xβ(t, u)=Λ− e 1 e− i χ( u ), (2.1.13) P P | | 11 which acts on vectors f(t)=(f0(t), f1(t), ...) of functions, as follows

∞ i n 1 X 1 tiu β − 1 β(t, u)f(t) n = e u fn 1(t β[u− ]). P | | − − For the sake of these arguments,
it is convenient
to introduce the following vector βJ(i) βJ(i) Virasoro generators: k (t) := ( k,n(t, n))n Z. ∈ k ∂ k+1 Proposition 2.2 The multiplication operator z and the differential operators ∂z z with z C∗, acting on the vertex operator X (t, z), have realizations as commutators, ∈ β βJ(1) βJ(2) in terms of the Heisenberg and Virasoro generators k (t, n) and k (t, n) : kX βJ(1) X z β(t, z) = k (t), β(t, z) ∂ h i zk+1X (t, z) = βJ(2)(t), X (t, z) . (2.1.14) ∂z β k β h i Corollary 2.3 Given a weight ρ(z)dz on R satisfying (2.1.1), we have ∂ zk+1f(z)X (t, z)ρ(z)= a βJ(2) (t) b βJ(1) (t) , X (t, z)ρ(z) . (2.1.15) ∂z β i k+i − i k+i+1 β " i 0 # X≥   Proof: Using (2.1.13) in the last line, compute ∂ zk+1f(z)X (t, z)ρ(z) ∂z β ρ (z) ∂ = ′ f(z) zk+1X (t, z)ρ(z)+ ρ(z) zk+1f(z)X (t, z) ρ(z) β ∂z β  
∞ ∂ ∞ = b zk+i+1X (t, z) ρ(z)+ ρ(z) a zk+i+1X (t, z) − i β ∂z i β 0 ! 0 ! X X ∞ ∞ = b βJ(1) , X (t, z)ρ(z) + a βJ(2) , X (t, z)ρ(z) − i k+i+1 β i k+i β " 0 # " 0 # X X (2.1.16)

2 3 11For α C, define [α] := (α, α , α , ...) C∞. The operator Λ is the shift matrix, with zeroes ∈ 2 3 ∈ everywhere, except for 1’s just above the diagonal, i.e., (Λv)n = vn+1.

29 2, p.30 §

establishing (2.1.14).

Giving the weight ρE(u)du = ρ(u)IE(u)du, with ρ and E as before, define the integrated vector vertex operator

Yβ(t, ρE) := duρ(u)Xβ(t, u). (2.1.17) ZE and the vector operator

:= Dk Bk − Vk 2r ∂ := ck+1f(c ) a βJ(2) (t) b βJ(1) (t) , (2.1.18) i i ∂c − i k+i − i k+i+1 1 i i 0 X X≥   consisting of a c-dependent boundary part and a (t, n)-dependent Virasoro part . Bk Vk Proposition 2.4 The following commutation relation holds:

[ , Y (t, ρ )] = 0 (2.1.19) Dk β E Proof: Integrating both sides of (2.1.14) over E, one computes:

∂ 2r dz zk+1f(z)X (t, z)ρ(z) = ( 1)ick+1f(c )X (t, c )ρ(c ) ∂z β − i i β i i ZE 1
X 2r ∂ = ck+1f(c ) X (t, z)ρ(z)dz i i ∂c β 1 i E X Z = [ , Y (t, ρ )] , (2.1.20) Bk β E while on the other hand

dz a βJ(2) b βJ(1) , X (t, z)ρ(z) i k+i − i k+i+1 β E " i 0 # Z X≥  

βJ(2) βJ(1) X = ai k+i bi k+i+1 , dzρE (z) β(t, z) − R " i 0 # X≥   Z = [ , Y (t, ρ )] . (2.1.21) Vk β E Subtracting both expressions (2.1.19) and (2.1.20) yields

0 = [ , Y (t, ρ )] = [ , Y (t, ρ )] , Bk −Vk β E Dk β E concluding the proof of proposition 2.4.

30 2, p.31 §

Proposition 2.5 The column vector,

n ∞ i β tiz I(t) := ∆n(z) e 1 k ρ(zk)dzk En | | P Z k=1 !n 0 Y ≥ is a fixed point for the vertex operator Yβ(t, ρE): (see definition (2.1.17))

(Y (t, ρ )I) = I , n 1. (2.1.22) β E n n ≥

Proof: We have

n ∞ i β tiz In(t) = ∆n(z) e 1 k IE(zk)ρ(zk)dzk Rn | | P Z kY=1   ∞ i tiu β(n 1) = duρE(u)e 1 u − R | | Z P n 1 n 1 − β − ∞ i zk β tiz 1 ∆n 1(z) e 1 k ρE(zk)dzk Rn−1 − u | − | P Z kY=1 Yk=1   ∞ i tiu β(n 1) = duρE(u)e 1 u − R | | Z P −i n 1 ∞ u ∂ − ∞ i β 1 i ∂t β tiz e− i ∆n 1(z) e 1 k ρE(zk)dzk P Rn−1 | − | Z k=1 P Y−i  ∞ i ∞ u ∂ β(n 1) tiu β 1 i ∂t = duρE(u) u − e 1 e− i τn 1(t) R | | P − Z P

= Yβ(t, ρE)I(t) . (2.1.23) n  

Proof of Theorem 2.1: From proposition 2.4 it follows that for n 1, ≥ 0 = [ , (Y (t, ρ ))n] I Dk β E = Y (t, ρ )nI Y (t, ρ )n I. (2.1.24) Dk β E − β E Dk −i i u ∂ tiu β i ∂t Taking the nth component for n 1 and k 1, and setting Xβ(t, u)= e e− i , ≥ ≥ − P and using (2.1.21) P

0 = ( I Y (t, ρ )n I) Dk − β E Dk n β n 1 = ( I) duρ (u)X (t; u)( u ) − ... duρ (u)X (t; u)( I) Dk n − E β | | E β Dk 0 Z Z = ( I) . Dk n 31 2, p.32 §

β (2) β (1) (0) Indeed ( kI)0 = 0 for k 1, since τ0 = 1 and k involves Jk , Jk and Jk for k 1: D ≥ − D ≥ − β (2) Jk is pure differentiation for k 1; β (1) ≥ −  Jk is pure differentiation, except for k = 1; but β (1) −  J 1 appears with coefficient nβ, which vanishes for n = 0;  (0)− J appears with coefficient n((n 1) β + 1), vanishing for n = 0. k − 2   2.1.3 Examples Example 1 : Gaussian β-integrals

The weight and the ai and bi, as in (2.1.1), are given by

V (z) z2 ρ(z)= e− = e− , V ′ = g/f =2z,

a0 =1, b0 =0, b1 =2, and all other ai, bi =0. From (2.1.5), the integrals

n 2 ∞ i β z + tiz In = ∆n(z) e− k i=1 k dzk (2.1.25) En | | P Z kY=1 satisfy the Virasoro constraints, for k 1, ≥ − 2r ∂ I = ck+2 I = βJ(2) a βJ(1) + βJ(1) I . (2.1.26) −Bk n − i ∂c n − k+1,n − k+1,n k+2,n n 1 i X   Introducing the following notation i +1 i +1 σ =(n )β + i +1 b =(n )β + i +1, (2.1.27) i − 2 − 0 − 2

the first three constraints have the following form, upon setting Fn = log In,

∂ ∂ ∂ ∂ n −1F = 2 iti F nt1 , 0F = 2 iti F σ1 −B  ∂t − ∂t −  − −B  ∂t − ∂t  − 2 1 ≥ i 1 2 ≥ i Xi 2 Xi 1     ∂ ∂ ∂ F = 2 σ it F. (2.1.28) −B1  ∂t − 1 ∂t − i ∂t  3 1 ≥ i+1 Xi 1   For later use, take the linear combinations

1 1 1 σ1 1 = 1, 2 = 0, 3 = 1 + 1 , (2.1.29) D −2B− D −2B D −2 B 2 B−   ∂F such that each i contains the pure term ∂F/∂ti, i.e., iF = + ... D D ∂ti 32 2, p.33 §

Example 2 (Laguerre β-integrals)

Here, the weight and the ai and bi, as in (2.1.1), are given by

V a z g z a e− = z e− , V ′ = = − , f z a =0, a =1, b = a, b =1, and all other a , b =0. 0 1 0 − 1 i i Thus from (2.1.4), the integrals

n ∞ i β a zk+ i=1 tizk In = ∆n(z) zk e− dzk (2.1.30) En | | P Z kY=1 satisfy the Virasoro constraints, for k 1, ≥ − 2r ∂ I = ck+2 I = βJ(2) a βJ(1) + βJ(1) I . (2.1.31) −Bk n − i ∂c n − k+1,n − k+1,n k+2,n n 1 i X   Introducing the following notation i +1 i +1 σ = =(n )β + i +1 b =(n )β + i +1+ a, i − 2 − 0 − 2 the first three have the form, upon setting F = Fn = log In,

∂ ∂ n − F = it F (σ + a) −B 1 ∂t − i ∂t  − 2 1 1 ≥ i Xi 1   ∂ ∂ ∂ F = σ it F −B0 ∂t − 1 ∂t − i ∂t  2 1 ≥ i+1 Xi 1   ∂ ∂ ∂ β ∂2 β ∂F 2 F = σ it F . −B1 ∂t − 2 ∂t − i ∂t − 2 ∂t2  − 2 ∂t 3 2 ≥ i+2 1 1 Xi 1     ∂F Again, replace the operators i by linear combinations i, such that iF = + ..., B D D ∂ti

1 = 1, 2 = 0 σ1 1, 3 = 1 σ2 0 σ1σ2 1. (2.1.32) D −B− D −B − B− D −B − B − B− Example 3 (Jacobi β-integral) This case is particularly important, because it covers not only the first integral, but also the second integral in the list (2.0.1), used in the problem of random permutations. The weight and the ai and bi, as in (2.1.1), are given by

V a b g a b +(a + b)z ρ(z) := e− = (1 z) (1 + z) ,V ′ = = − − f 1 z2 − 33 2, p.34 §

a =1, a =0, a = 1, b = a b, b = a + b, and all other a , b =0. 0 1 2 − 0 − 1 i i The integrals

n ∞ i β a b tiz In = ∆n(z) (1 zk) (1 + zk) e i=1 k dzk (2.1.33) En | | − P Z kY=1 satisfy the Virasoro constraints (k 1): ≥ − 2r ∂ I = ck+1(1 c2) I − Bk n − i − i ∂c n 1 i X = βJ(2) βJ(2) + b βJ(1) + b βJ(1) I . (2.1.34) k+2,n − k,n 0 k+1,n 1 k+2,n n   Introducing σ =(n i+1 )β + i +1+ b , the first four have the form (k = 1, 0, 1, 2) i − 2 1 − ∂ ∂ ∂ −1F = σ1 + iti iti F + n(b0 t1) −B  ∂t ∂t − ∂t −  − 1 ≥ i+1 ≥ i 1 Xi 1 Xi 2   ∂ ∂ ∂ ∂ β ∂2 β ∂F 2 n F = σ + b + it ( )+ F + (σ b ) −B0  2 ∂t 0 ∂t i ∂t − ∂t 2 ∂t2  2 ∂t − 2 1 − 1 2 1 ≥ i+2 i 1 1 Xi 1     ∂ ∂ ∂ ∂ ∂ ∂2 F = σ + b (σ b ) + it ( )+ β F −B1  3 ∂t 0 ∂t − 1 − 1 ∂t i ∂t − ∂t ∂t ∂t  3 2 1 ≥ i+3 i+1 1 2 Xi 1  ∂F ∂F  +β ∂t1 ∂t2 ∂ ∂ ∂ ∂ ∂ F = σ + b (σ b ) + it ( ) −B2  4 ∂t 0 ∂t − 2 − 1 ∂t i ∂t − ∂t 4 3 2 ≥ i+4 i+2 Xi 1 β ∂2 ∂2 ∂2 β ∂F ∂F ∂F ∂F + ( +2 ) F + ( )2 ( )2 +2 . 2 ∂t2 − ∂t2 ∂t ∂t 2 ∂t − ∂t ∂t ∂t 2 1 1 3   2 1 1 3  (2.1.35)

2.2 Double matrix integrals Consider now weights of the form

i j ρ(x, y)= e i,j≥1 rij x y ρ(x)˜ρ(y), (2.2.1) P defined on a product of intervals F F R2, with rational logarithmic derivative 1 × 2 ⊂ i ˜ i ρ′ g i 0 bix ρ˜′ g˜ i 0 biy = = ≥ and = = ≥ , i ˜ i − ρ f i 0 aix − ρ˜ f i 0 a˜iy P ≥ P ≥ P 34 P 2, p.35 §

satisfying

lim f(x)ρ(x)xk = lim f˜(y)˜ρ(y)yk = 0 for all k 0. (2.2.2) x ∂F y ∂F˜ ≥ → → Consider subsets of the form

r s 2 E = E1 E2 := i=1[c2i 1,c2i] i=1[˜c2i 1, c˜2i] F1 F2 R . (2.2.3) × ∪ − × ∪ − ⊂ × ⊂ A natural deformation of the second integral in the list (2.0.1) is given by the following integrals:

n ∞ i i (tix siy ) In(t,s,r; E)= ∆n(x)∆n(y) e i=1 k− k ρ(xk,yk)dxkdyk (2.2.4) En P Z Z Yk=1 J(i) J˜(i) In the theorem below, k,n and k,n are vectors of operators, whose components are given by the operators (2.1.10) for β = 1; i.e.,

J(i) βJ(i) J˜(i) βJ(i) k,n(t)= k,n(t) , k,n(s) := k,n(t) ; β=1 β=1, t s 7→− thus, from (2.1.10) and (2.1.11), one finds: 1 J(2) (t)= J (2)(t)+(2n + k + 1)J (1)(t)+ n(n + 1)J (0) , (2.2.5) k,n 2 k k k   satisfying the Heisenberg and Virasoro relations (2.1.8), with central charge c = 2 and − c′ =1/2. The ai, a˜i, bi, ˜bi,ci, c˜i,rij given by (2.2.1), (2.2.2) and (2.2.3) define differential oper- ators:

2r ∂ ∂ k+1 J(2) J(1) k,n : = ci f(ci) ai( k+i,n + mrmℓ ) bi k+i+1,n D ∂ci − ∂rm+k+i,ℓ − 1 i 0 m,ℓ 1 X X≥  X≥  2r ∂ ∂ ˜ k+1 ˜ J˜(2) ˜ J˜(1) k,n : = c˜i f(˜ci) a˜i( k+i,n + ℓrmℓ ) bi k+i+1,n . D ∂c˜i − ∂rm,ℓ+k+i − 1 i 0 m,ℓ 1 X X≥  X≥  (2.2.6)

Theorem 2.6 (Adler-van Moerbeke [3, 4]) Given ρ(x, y) as in (2.2.1), the integrals

n ∞ i i (tix siy ) In(t,s,r; E) := ∆n(x)∆n(y) e i=1 k− k ρ(xk,yk)dxkdyk (2.2.7) En P ZZ Yk=1 satisfy two families of Virasoro equations for k 1: ≥ − I (t,s,r; E)=0 and ˜ I (t,s,r; E)=0. (2.2.8) Dk,n n Dk,n n

35 2, p.36 §

The proof of this statement is very similar to the one for β-integrals. Namely, define the vector vertex operator,

−i −i ∞ i i ∞ u ∂ v ∂ 1 (tiu siv ) 1 ( i ∂t i ∂s ) X12(t, s; u, v)=Λ− e 1 − e− i − i χ(uv), (2.2.9) P P J(i) which, as a consequence of Proposition 2.2, interacts with the operators k (t) = J(i) k,n(t, n) , as follows: n Z   ∈ kX J(1) X u 12(t, s; u, v) = k (t), 12(t, s; u, v) ∂ h i uk+1X (t, s; u, v) = J(2)(t), X (t, s; u, v) . (2.2.10) ∂u 12 k 12 h i k ∂ k+1 k A similar statement can be made, upon replacing the operators u and ∂u u by v ∂ k+1 J˜(i) and ∂v v , and upon using the k (s)’s. Finally, one checks that the integral vertex operator

Y(t, s; ρE) := dxdyρ(x, y)X12(t, s; x, y) (2.2.11) Z ZE

commutes with the two vectors of differential operators k =( k,n)n Z, as in (2.2.6): D D ∈ , Y(t, s; ρ ) = ˜ , Y(t, s; ρ ) =0, Dk E Dk E h i h i and that the vector I =(I0 =1,I1, ...) of integrals (2.2.7) is a fixed point for Y(t, s; ρE),

Y(t, s; ρE)I(t,s,r; E)= I(t,s,r; E).

Then, as before, the proof of Theorem 2.6 hinges ultimately on the fact that Dk,0 annihilates I0 = 1.

2.3 Integrals over the unit circle We now deal with the fourth type of integral in the list (2.0.1), which we deform, this time, by inserting two sequences of times t1, t2, ... and s1,s2, .... The following theorem holds:

Theorem 2.7 (Adler-van Moerbeke [7]) The multiple integrals over the unit circle S1,

n ∞ i −i 2 (tiz siz ) dzk In(t, s)= ∆n(z) e 1 k− k , n> 0, (2.3.1) (S1)n | | P 2πizk Z Yk=1

36 2, p.37 §

with I0 =1, satisfy an SL(2,Z)-algebra of Virasoro constraints:

k = 1, θ =0 − θ I (t, s)=0, for k =0, θ arbitrary only, (2.3.2) Dk,n n    k =1, θ =1  where the operators θ := θ (t, s, n), k Z, n 0 are given by Dk,n Dk,n ∈ ≥ θ J(2) J(2) J(1) J(1) k,n := k,n(t, n) k,n( s, n) k θ k,n(t, n)+(1 θ) k,n( s, n) , (2.3.3) D − − − − − − −   J(i) βJ(i) with k,n(t, n) := k,n(t, n) , as in (2.1.11). β=1

The explicit expressions are

∂ ∂ ∂ − I = (i + 1)t (i 1)s − + n t + I =0 D 1 n  i+1 ∂t − − i 1 ∂s 1 ∂s  n ≥ i ≥ i 1 Xi 1 Xi 2    ∂ ∂  I = it is I =0 (2.3.4) D0 n i ∂t − i ∂s n ≥ i i Xi 1   ∂ ∂ ∂ I = (i + 1)s + (i 1)t − + n s + I =0. D1 n − i+1 ∂s − i 1 ∂t 1 ∂t  n ≥ i ≥ i 1 Xi 1 Xi 2    

Here the key vertex operator is a reduction of X12(t, s; u, v), defined in the pre- Z θ vious section (formula (2.2.9)). For all k , the vector of operators k(t, s) = θ (t, s, n) form a realization of the first∈ order differential operators Dd uk+1, us- Dk,n n Z du ing the vertex∈ operator X (t, s; u,u 1), namely
12 − d X (t, s; u,u 1) X (t, s; u,u 1) uk+1 12 − = θ(t, s), 12 − . (2.3.5) du u Dk u   Indeed, d k 1 u u X (t, s; u,u− ) du 12

∂ k+1 ∂ 1 k k k = u v − kθu k(1 θ)v− X (t, s; u, v) ∂u − ∂v − − − 12   v= u − J(2) J(2) J(1) J(1) X = k (t) k( s) k θ k (t)+(1 θ) k ( s) , 12(t, s; u, u) − − − − − − − θ 1 = h (t, s), X (t, s; u,u− .  i Dk 12 The θ := θ(t, s) satisfy Virasoro relations with central charge = 0, Dk Dk θ, θ =(k ℓ) θ , (2.3.6) Dk Dℓ − Dk+ℓ   37 3, p.38 § and, from (2.3.5) the following commutation relation holds:

θ du 1 k(t, s), Y(t, s) =0, with Y(t, s) := X12(t, s; u,u− ). (2.3.7) D 1 2πiu ZS   The point is that the column vector I(t, s)=(I0,I1, ...) of integrals (2.3.1), is a fixed point for Y(t, s):

(Y(t, s)I) = I , n 1, (2.3.8) n n ≥ which is shown in a way, similar to Proposition 2.5. Proof of Theorem 2.7: Here again the proof is similar to the one of Theorem 2.1. Taking the nth component and the nth power of Y(t, s), with n 1, and noticing from θ ≥ the explicit formulae (2.3.4) that k(t, s)I 0 = 0, we have, by means of a calculation similar to the proof of Theorem 2.1,D
0 = [ θ, Y(t, s)n]I Dk n = θY(t, s)nI Y(t, s)n θI Dk −
Dk n = θI Y(t, s)n θI = θI . Dk − Dk n D
k n

3 Integrable systems and associated matrix inte- grals

3.1 Toda lattice and Hermitian matrix integrals 3.1.1 Toda lattice, factorization of symmetric matrices and orthogonal poly- nomials

V (z) Given a weight ρ(z)= e− defined as in (2.1.1), the inner-product over E R, ⊆

i f,g = f(z)g(z)ρ (z)dz, with ρ (z) := e tiz ρ(z), (3.1.1) h it t t ZE P leads to a moment matrix

i j mn(t)=(µij(t))0 i,j

∂µij ∂m k commuting = µi+k,j, and thus ∞ =Λ m . (3.1.3) ∂t ∂t ∞ vector fields k k   12 H¨ankel means µij depends on i + j only

38 3, p.39 §

Another important ingredient is the factorization of m into a lower- times an upper- ∞ triangular matrix13 1 1 m (t)= S(t)− S(t)⊤− , S(t) = lower triangular with ∞ non-zero diagonal elements. The main ideas of the following theorem can be found in [2, 5]. Remember c = (c1, ..., c2r) denotes the boundary points of the set E. dM refers to properly normalized Haar measure on . Hn Theorem 3.1 The determinants of the moment matrices n 1 2 τn(t, c) := det mn(t, c) = ∆n(z) ρt(zk)dzk n! En Z kY=1 tr( V (M)+ ∞ t M i) = e − 1 i dM, (3.1.4) n(E) P ZH satisfy (i) Virasoro constraints (2.1.7) for β =2, 2r ∂ ck+1f(c ) + a J(2) b J(1) τ (t, c)=0. (3.1.5) − i i ∂c i k+i,n − i k+i+1,n n 1 i i 0 ! X X≥   (ii) The KP-hierarchy14(k =0, 1, 2,... ) ∂ 1 ∂ 1 ∂ 1 ∂2 s , , ,... τ τ =0, k+4 ∂t 2 ∂t 3 ∂t − 2 ∂t ∂t n ◦ n  1 2 3 1 k+3  of which the first equation reads:
∂ 4 ∂ 2 ∂2 ∂2 2 +3 4 log τ +6 log τ =0. (3.1.6) ∂t ∂t − ∂t ∂t n ∂t2 n  1   2  1 3 !  1  (iii) The standard Toda lattice, i.e., the symmetric tridiagonal matrix 1/2 ∂ τ1 τ0τ2 log 2 0 ∂t1 τ0 τ1  1/2 1/2  τ0τ2 ∂  τ2 τ1τ3 1 τ 2 ∂t log τ τ 2 L(t) := S(t)ΛS(t)− = 1 1 1 2 (3.1.7)  1/2     τ1τ3 ∂  τ3   0 τ 2 ∂t log τ   2 1 2   .     ..    13   This factorization is possible for those t’s for which τn(t) := det mn(t) = 0 for all n> 0. 14 6 Given a polynomial p(t1,t2, ...), define the customary Hirota symbol p(∂t)f g := ∞◦ i ∂ ∂ i p( , , ...)f(t + y)g(t y) . The s ’s are the elementary Schur polynomials e 1 t z := ∂y1 ∂y2 ℓ − y=0 P s (t)zi and s (∂˜) := s ( ∂ , 1 ∂ ,... ). i≥0 i ℓ ℓ ∂t 1 2 ∂t2

P 39 3, p.40 §

satisfies the commuting equations15

∂L 1 k = (L )s, L . (3.1.8) ∂t 2 k   (iv) Eigenvectors of L: The tridiagonal matrix L admits two independent eigenvectors:

p(t; z)=(pn(t; z))n 0 satisfying (L(t)p(t; z))n = zpn(t; z), n 0, where • ≥ ≥ pn(t; z) are nth degree polynomials in z, depending on t C∞, orthonormal with respect to the t-dependent inner product16 (3.1.1) ∈

p (t; z),p (t; z) = δ ; h k ℓ it kℓ they are eigenvectors of L, i.e., L(t)p(t; z)= zp(t; z), and enjoy the following 2 representations: (χ(z)=(1,z,z , ...)⊤)

pn(t; z) := (S(t)χ(z))n 1 m (t) z 1  n  = det . τn(t)τn+1(t) .  n   µn,0 ... µn,n 1 z  p  −  1  n 1/2 τn(t [z− ]) τn+1(t) = z hn− − , hn := (3.1.9) τn(t) τn(t)

pn(t;u) q(t, z)=(qn(t; z))n 0, with qn(t; z) := z Rn ρt(u)du, satisfying (L(t)q(t; z))n = • ≥ z u zq (t; z), n 1; q (t; z) enjoys the following− representations: n ≥ n R

pn(t; u) 1 1 ⊤− − qn(t; z) := z ρt(u)du = S (t)χ(z ) n Rn z u Z − 1 = S(t)m (t)χ(z−
) ∞ n 1 n 1/2 τn+1(t + [
z− ]) = z− hn− . τn(t) (3.1.10)

In the case β = 2, the Virasoro generators (2.1.11) take on a particularly elegant form, namely for n 0, ≥ J(2) J(1) J(1) (2) (1) 2 k,n(t) = : i,n(t) j,n(t): = Jk (t)+2nJk (t)+ n δ0k i+Xj=k J(1) (1) k,n(t) = Jk (t)+ nδ0k, 15 ()s means: take the skew-symmetric part of (); for more details, see subsection 3.1.2. 16The explicit dependence on the boundary points c will be omitted in this point (iv).

40 3, p.41 §

with17

2 (1) ∂ 1 (2) ∂ ∂ 1 Jk = + ( k)t k , Jk = + iti + itijtj. (3.1.11) ∂tk 2 − − ∂ti∂tj ∂tj 4 i+j=k i+j=k i j=k X −X −X− Statement (i) is already contained in section 2, whereas statement (ii) will be established in subsection 3.1.2, using elementary methods.

3.1.2 Sketch of Proof Orthogonal polynomials and τ-function representation: The representation (3.1.4) of the determinants of moment matrices as integrals follows immediately from the fact that the square of a Vandermonde determinant can be represented as a sum of determinants 2 ℓ+k 2 ∆ (u1,...,un)= det uσ(k)− . 1 k,ℓ n σ Sn ≤ ≤ X∈   Indeed,

n!τn(t) = n! det mn(t)

ℓ+k 2 = det zσ(k)− ρt(zσ(k))dzσ(k) E 1 k,ℓ n σ Sn   X∈ Z ≤ ≤ ℓ+k 2 = det zσ(k)− ρt(zσ(k))dzσ(k) En 1 k,ℓ n σ Sn ≤ ≤ ∈ Z   X n 2 = ∆n(z) ρt(zk)dzk, En Z kY=1 whereas the representation (3.1.4) in terms of integrals over Hermitian matrices follows from section 1.1. The Borel factorization of m is responsible for the orthonormality of the polyno- ∞ mials pn(t; z)=(S(t)χ(z))n; indeed,

pk(t; z),pℓ(t; z) 0 k,ℓ< = Sχ(z)(Sχ(z))⊤ρt(z)dz = Sm S⊤ = I. h i ≤ ∞ ∞ ZE

Note that Sχ(z)(Sχ(z))⊤ should be viewed as a semi-infinite matrix obtained by multi- plying a semi-infinite column and row. The determinantal representation (3.1.9) follows k at once from noticing that pn(t; z), z = 0 for 0 k n 1, because taking that inner-product produces twoh identical columnsi in the≤ matrix≤ − thus obtained. From the 1 n 1/2 same representation (3.1.9), one has pn(t; z)= hn− z + ... , where hn := (τn+1/τn(t)) .

17 (1) The expression Jk = 0 for k = 0.

41 3, p.42 §

The “Sato” representation (3.1.9) of pn(t; z) in terms of the determinant τn(t) of the moment matrix can be shown by first proving the Heine representation of the orthogonal polynomials, which goes as follows: hnpn(t; z)

1 1 mn(t) z = det  .  τn .  n   µn,0 ... µn,n 1 z   −   0 1 n 1 u1 u2 un− 1 1 2 ··· n u1 u2 un z n 1  . ··· . .  = det . . . ρt(ui)dui τ n n E  n 1 n 2n 2 n 1  1 Z  u1− u2 un − z −  Y  n n+1 ··· 2n 1 n   u u u − z   1 2 ··· n   0 0 0  u1 u2 un 1 1 1 ··· 1 u1 u2 un z n 1  . ··· . .  0 1 n 1 = det . . . u1u2 un− ρt(ui)dui τ n ··· n E  n 1 n 1 n 1 n 1  1 Z  u1− u2− un− z −   ···  Y  un un un zn   1 2 ··· n   0 0 0  uσ(1) uσ(2) uσ(n) 1 1 1 ··· 1 uσ(1) uσ(2) uσ(n) z 1  . ··· . .  = det . . . τn En  n 1 n 1 n 1 n 1  Z  u − u − u − z −   σ(1) σ(2) ··· σ(n)   un un un zn   σ(1) σ(2) ··· σ(n)   n  0 1 n 1 u u u − ρ (u )du , σ(1) σ(2) ··· σ(n) t σ(i) σ(i) 1 Y for any permutation σ Sn 0 0 0 ∈ u1 u2 un 1 1 1 ··· 1 u1 u2 un z 1  . ··· . .  = det . . . τn En  n 1 n 1 n 1 n 1  Z  u1− u2− un− z −   ···   un un un zn   1 2 ··· n   n  σ 0 1 n 1 ( 1) u u u − ρ (u )du − σ(1) σ(2) ··· σ(n) t i i 1 n Y 1 2 = ∆n(u) (z uk)ρt(uk)duk, upon summing over all σ. n!τn En − Z Yk=1

42 3, p.43 §

Therefore, using again the representation of ∆2(z) as a sum of determinants, Heine’s formula leads to

hnpn(t, z) n n z ℓ+k 2 uσ(k) = det uσ(k)− 1 ρt(uσ(k))duσ(k), n!τn En 1 k,ℓ n − z σ Sn ≤ ≤ k=1 Z ∈     n X Y z ℓ+k 2 1 ℓ+k 1 = det uσ(k)− uσ(k)− ρt(uσ(k))duσ(k) n!τn En − z 1 k,ℓ n σ Sn   Z X∈ ≤ ≤ zn 1 = det µ µ τ ij − z i,j+1 n  0 i,j n 1 n ≤ ≤ − z 1 = det µij(t [z− ]) 0 i,j n 1 τn − ≤ ≤ − 1 n τn(t [z− ])
= z − , (3.1.12) τn(t) invoking the fact that

−i ∞ z i u 1 i+j 1 ti i u i+j µij(t [z− ]) = u e − ρ(u)du = u 1 ρ(u)du − P   − z Z Z 1  = µ µ . i+j − z i+j+1 Formula (3.1.10) follows from computing on the one hand S(t)m χ(z) using the ∞ explicit moments µij, together with (3.1.12), and on the other hand the equivalent 1 1 n k expression S⊤− (t)χ(z− ). Indeed, using (S(t)χ(z))n = pn(t; z)= 0 pnk(t)z ,

j j P (Sm ) z− = z− pnℓ(t)µℓj ∞ nj j 0 j 0 ℓ 0 X≥ X≥ X≥ j ℓ+j = z− pnℓ(t) u ρt(u)du j 0 ℓ 0 E X≥ X≥ Z u j = p (t)uℓ ρ (u)du nℓ z t E ℓ 0 j 0 Z X≥ X≥   p (t, u)ρ (u) = z n t du. z u ZE −

Mimicking computation (3.1.12), one shows

1 1 j τn+1(t + [z− ]) n h S⊤− (t) z− = z− , n nj τ (t) j 0 n X≥

43 3, p.44 §

1 from which (3.1.10) follows, upon using Sm = S⊤− . Details of this and subsequent ∞ derivation can be found in [5, 6]. 2 The vectors p and q are eigenvectors of L. Indeed, remembering χ(z)=(1,z,z , ...)⊤, and the shift (Λv)n = vn+1, we have

1 1 Λχ(z)= zχ(z) and Λ⊤χ(z− )= zχ(z− ) ze , with e = (1, 0, 0, ...)⊤. − 1 1 1 1 Therefore, p(z)= Sχ(z) and q(z)= S⊤− χ(z− ) are eigenvectors, in the sense

1 Lp = SΛS− Sχ(z)= zSχ(z)= zp 1 1 1 1 1 1 1 L⊤q = S⊤− Λ⊤S⊤S⊤− χ(z− )= zS⊤− χ(z− ) zS⊤− e = zq zS⊤− e . − 1 − 1 Then, using L = L⊤, one is lead to ((L zI)p) =0, for n 0 and ((L zI)q) =0, for n 1. − n ≥ − n ≥ Toda lattice and Lie algebra splitting: The Lie algebra splitting of semi-infinite matrices and the corresponding projections (used in (3.1.8)), denoted by ( )s and ( )b are defined as follows: s = skew-symmetric matrices gl( )= s b { } . ∞ ⊕ b = lower-triangular matrices  { } Conjugating the shift matrix Λ by S(t) yields a matrix

1 L(t) = S(t)ΛS(t)− 1 1 = SΛS− S⊤− S⊤

= SΛm S⊤, using (3.1.3), ∞ = Sm Λ⊤S⊤, using Λm = m Λ⊤, ∞ ∞ ∞ 1 1 = S(S− S⊤− )Λ⊤S⊤, using again (3.1.3), 1 = (SΛS− )⊤ = L(t)⊤, which is symmetric and thus tridiagonal. Moreover, from (3.1.3) one computes

k ∂m 0 = S Λ m ∞ S⊤ ∞ − ∂t  k  k 1 ∂ 1 1 = SΛ S− S (S− S⊤− )S⊤ − ∂tk

k ∂S 1 1 ∂S⊤ = L + S− + S⊤− . ∂tk ∂tk

Upon taking the () and ()0 parts of this equation (A means the lower-triangular part − − of the matrix A, including the diagonal and A0 the diagonal part) leads to

k ∂S 1 1 ∂S⊤ ∂S 1 1 k (L ) + S− + S⊤− = 0 and S− = (L )0. − ∂t ∂t ∂t −2 k  k 0  k 0 44 3, p.45 §

Upon observing that for any symmetric matrix

a c a 0 a c a c = =2 , c b b 2c b c b − c b 0      −   it follows that the matrices L(t), S(t) and the vector p(t; z)=(pn(t; z))n 0 = S(t)χ(z) ≥ satisfy the (commuting) differential equations and the eigenvalue problem

∂S 1 k = (L )bS, L(t)p(t; z)= zp(t; z), (3.1.13) ∂tk −2 and thus ∂L 1 k ∂p 1 k = (L )b, L , = (L )bp. ∂t − 2 ∂t −2 k   k (Standard Toda lattice) The bilinear identity: The functions τ (t) satisfy the following identity, for n n ≥ m +1, t, t′ C, where one integrates along a small circle about , ∈ ∞

′ i 1 1 (ti t )z n m 1 τn(t [z− ])τm+1(t′ + [z− ])e − i z − − dz =0. (3.1.14) z= − P I ∞ An elementary proof can be given by expressing the left hand side of (3.1.14), in terms of pn(t; z) and pm(t, z), using (3.1.9) and (3.1.10). One uses below the following identity (see [2])

g(u) f(z)g(z)dz = f, du , (3.1.15) R R z u Z  Z − ∞ 18 involving the residue pairing . So, modulo terms depending on t and t′ only, the left hand side of (3.1.14) equals

∞ ′ i ∞ ′ i n (ti t )z n m 1 m+1 pm(t′; u) t u dz z− pn(t; z)e 1 − i z − − z e 1 i ρ(u)du z= P R z u P I ∞ Z − ′ i ′ i (ti t )z t z = pn(t; z)e − i pm(t′; z)e i ρ(z)dz, using (3.1.15), R Z P P i tiz = pn(t; z)pm(t′; z)e ρ(z)dz =0, when m n 1. (3.1.16) R P ≤ − 18 Z i + −j−1 The residue pairing about z = between f = i≥0 aiz and g = j∈Z bj z is defined as: ∞ ∈ H ∈ H Pdz P f,g ∞ = f(z)g(z) = a b . h i 2πi i i z=∞ ≥ I Xi 0

45 3, p.46 §

The KP-hierarchy: Setting n = m + 1, shifting t t y, t′ t + y, evaluating the 7→ − 7→ residue and Taylor expanding in yk and using the Schur polynomials sn, leads to (see footnote 14 for the definition of p(∂ )f g.) t ◦ 1 ∞ 2y zi 1 1 0 = dz e− 1 i τ (t y [z− ])τ (t + y + [z− ]) 2πi n − − n I P 1 ∞ ∞ ∞ ∂ i j 1 yk ∂t = dz z si( 2y) z− sj(∂˜) e k τn τn 2πi − P ◦ 0 ! 0 ! I X X ∞ ∂ ∞ 1 yk ∂t = e k si( 2y)si+1(∂˜)τn τn P − ◦ 0 X ∞ ∂ ∂ ∞ = 1+ y + O(y2) + s (∂˜)( 2y + O(y2)) τ τ j ∂t ∂t i+1 − i n ◦ n 1 j ! 1 1 ! X X ∂ ∞ ∂ ∂ = + y 2s (∂˜) τ τ + O(y2), ∂t k ∂t ∂t − k+1 n ◦ n 1 1 k 1 ! X   thus yielding (3.1.6), taking into acount the fact that (∂/∂t )τ τ = 0 and the coefficient 1 ◦ of yk is trivial for k =1, 2. The Riemann-Hilbert problem: Observe that, as a function of z, the integral (3.1.10) has a jump accross the real axis

1 pn(t; u) 1 pn(t; u) lim ρt(u)du = pn(t, z)ρt(z)+ lim ρt(u)du, ′ ′ 2πi z →z R z′ u 2πi z →z R z′ u Iz′<0 Z − Iz′>0 Z − and thus we have: (see [29, 19, 5]) Corollary 3.2 The matrix

−1 −1 τn(t [z ]) n τn+1(t+[z ]) n 1 − z z− − τn(t) τn(t) Yn(z)=  −1 −1  τn−1(t [z ]) n 1 τn(t+[z ]) n − z − z−  τn(t) τn(t)    satisfies the Riemann–Hilbert problem19:

1. Yn(z) holomorphic on the C+ and C − 1 2πiρt(z) 2. Yn (z)= Yn+(z) (Jump condition) −   0 1 n   z− 0 1 3. Y ′(z) =1+ O(z− ), when z . n   0 zn →∞

19   C+ and C− denote the Siegel upper and lower half plane.

46 3, p.47 §

3.2 Pfaff Lattice and symmetric/symplectic matrix integrals 3.2.1 Pfaff lattice, factorization of skew-symmetric matrices and skew- orthogonal polynomials Consider an inner-product, with a skew-symmetric weightρ ˜(y, z),

i i ti(y +z ) f,g t = f(y)g(z)e ρ˜(y, z)dy dz, withρ ˜(z,y)= ρ˜(y, z). (3.2.1) h i R2 − Z Z P Since f,g = g, f , the moment matrix, depending on t =(t , t ,... ), h it −h it 1 2 i j mn(t)=(µij(t))0 i,j n 1 =( y , z t)0 i,j n 1 ≤ ≤ − h i ≤ ≤ − is skew-symmetric. It is clear from formula (3.2.1) that the semi-infinite matrix m ∞ evolves in t according to the commuting vector fields:

∂µij ∂m k k = µi+k,j + µi,j+k, i.e., ∞ =Λ m + m Λ⊤ . (3.2.2) ∂tk ∂tk ∞ ∞ Since m is skew-symmetric, m does not admit a Borel factorization in the stan- ∞ ∞ dard sense, but m admits a unique factorization, with an inserted semi-infinite, skew- ∞ symmetric matrix J, with J 2 = I, of the form (1.1.12): (see [2]) − 1 1 m (t)= Q− (t)J Q⊤− (t), ∞ where . .. 0  0  Q 0  2n,2n   0 Q  Q(t)=  2n,2n  K. (3.2.3)   ∈  Q2n+2,2n+2 0     ∗ 0 Q2n+2,2n+2     .   ..      K is the group of lower-triangular invertible matrices of the form above, with Lie algebra k. Consider the Lie algebra splitting, given by

k = lower-triangular matrices of the form (3.2.3) gl( )= k n { } ∞ ⊕ n = sp( )= a such that Ja⊤J = a ,  ∞ { }

47 3, p.48 §

(3.2.4) with unique decomposition20

a = (a)k +(a)n 1 = (a J(a+)⊤J)+ (a0 J(a0)⊤J) − − 2 −   1 + (a + J(a )⊤J)+ (a + J(a )⊤J) . (3.2.5) + + 2 0 0   Consider as a special skew-symmetric weight (3.2.1): (see [13])

α V˜ (y) ρ˜(y, z)=2D δ(y z)˜ρ(y)˜ρ(z) , with α = 1, ρ˜(y)= e− , (3.2.6) − ∓ together with the associated inner-product21 of type (3.2.1):

i i ti(y +z ) α f,g t = f(y)g(z)e 2D δ(y z)˜ρ(y)˜ρ(z)dy dz (3.2.7) h i R2 − Z Z P ∞ t (yi+zi) f(y)g(z)e 1 i ε(y z)˜ρ(y)˜ρ(z)dy dz, for α = 1 R2 P − − =  ZZ   ∞ 2t yi 2  f,g (y)e 1 i ρ˜(y) dy, for α = +1 R{ } P  Z  in terms of the Wronskian f,g := ∂f g f ∂g . The moments with regard to these { } ∂y − ∂y inner-products (with that precise definition of time t!) satisfy the differential equations ∂µij/∂tk = µi+k,j + µi,j+k, as in (3.2.2). It is well known that the determinant of an odd skew-symmetric matrix equals 0, whereas the determinant of an even skew-symmetric matrix is the square of a polynomial in the entries, the Pfaffian 22. Define now the “Pfaffian τ-functions”, defined with regard

20 a± refers to projection onto strictly upper (strictly lower) triangular matrices, with all 2 2 × diagonal blocks equal zero. a0 refers to projection onto the “diagonal”, consisting of 2 2 blocks. 21ε(x) = sign x, having the property ε′ =2δ(x). × 22 with a sign specified below. So det(m2n−1(t)) = 0 and

1/2 (det m2n(t)) = pf(m2n(t)) n 1 −1 = (dx dx ... dx − ) µ (t)dx dx . n! 0 ∧ 1 ∧ ∧ 2n 1  ij i ∧ j  ≤ ≤ − 0 i

48 3, p.49 § to the inner-products (3.2.7) :

∞ i i pf ykzℓε(y z)e 1 ti(y +z )ρ˜(y)˜ρ(z)dydz ,α = 1 R2 − P 0 k,ℓ 2n 1 − τ2n(t) :=  ZZ  ≤ ≤ − ∞ i  k ℓ 1 2tiy 2  pf y ,y e ρ˜ (y)dy , α = +1 R{ } P 0 k,ℓ 2n 1 Z  ≤ ≤ −  (3.2.8)  Setting ρ˜(z)= ρ(z)I (z) for α = 1 E − ρ˜(z)= ρ1/2(z)I (z), t t/2 for α = +1,  E → in the identities (3.2.8) leads to the identities (3.2.9) between integrals and Pfaffians, spelled out in Theorem 3.3 below. Remember c =(c1, ..., c2r) stands for the boundary points of the disjoint union E R. (E) and (E) denotes the set of symmet- ⊂ Sn Tn ric/symplectic ensemble with spectrum in E.

Theorem 3.3 (Adler-Horozov-van Moerbeke [9], Adler-van Moerbeke [7]) The follow- ing integrals In(t, c) are Pfaffians:

n ∞ i β tiz In = ∆n(z) e 1 k ρ(zk)dzk En | | P Z kY=1   β =1 Tr( V (X)+ ∞ t Xi)  = e − 1 i dX, (n even)  n(E) P  ZS  k ℓ ∞ t (yi+zi)  = n!pf y z ε(y z)e 1 i ρ(y)ρ(z)dydz   E2 − P 0 k,ℓ n 1  ZZ  ≤ ≤ −  = n!τn(t, c)  =  (3.2.9)   β =4 ∞ i Tr( V (X)+ tiX )  = e − 1 dX, (n arbitrary)  2n(E) P  ZT  k ℓ ∞ t yi  = n!pf y ,y e 1 i ρ(y)dy  { }  E P 0 k,ℓ 2n 1  Z  ≤ ≤ −  = n!τ2n(t/2,c).   The In and τn’s satisfy (i) The Virasoro constraints23 (2.1.7) for β =1, 4,

2r ∂ ck+1f(c ) + a βJ(2) b βJ(1) I (t, c)=0. (3.2.10) − i i ∂c i k+i,n − i k+i+1,n n 1 i i 0 ! X X≥   ′ i 23 ρ biz i here the ai’s and bi’s are defined in the usual way, in terms of ρ(z); namely, ρ = aiz − P P 49 3, p.50 §

(ii) The Pfaff-KP hierarchy: (see footnote 14 for notation)

2 ˜ 1 ∂ ˜ sk+4(∂) τn τn = sk(∂) τn+2 τn 2 (3.2.11) − 2 ∂t ∂t ◦ ◦ −  1 k+3  n even, k =0, 1, 2, ... . of which the first equation reads (n even)

4 2 2 2 2 ∂ ∂ ∂ ∂ τn 2τn+2 +3 4 log τ +6 log τ = 12 − . ∂t ∂t − ∂t ∂t n ∂t2 n τ 2  1   2  1 3 !  1  n

(iii) The Pfaff Lattice: The time-dependent matrix L(t), zero above the first superdiag- onal, obtained by dressing up Λ,

1 ∗ (h /h )1/2 O  2 0  ∗ 1 ∗ 1  (h /h )1/2  L(t)= Q(t)ΛQ(t)− =  4 2  (3.2.12)  ∗     ∗ .   ..   ∗        satisfies the Hamiltonian commuting equations

∂L i = [ (L )k, L]. (Pfaff lattice) (3.2.13) ∂ti −

(iv) Skew-orthogonal polynomials: The vector of time-dependent polynomials q(t; z) := (qn(t; z))n 0 = Q(t)χ(z) in z satisfy the eigenvalue problem ≥ L(t)q(t, z)= zq(t, z) (3.2.14)

50 3, p.51 §

and enjoy the following representations (with h = τ2n+2(t) ) 2n τ2n(t) 1

1/2 m2n+1(t) z h2−n  .  q2n(t; z) = pf . τ2n(t)  2n   z     1 z . . . z2n 0   − − −   1  2n 1/2 τ2n(t [z− ]) 2n 1/2 = z h2−n − = z h2−n + ... , τ2n(t)

1 µ0,2n+1 m2n(t) z µ1,2n+1 1/2  . .  h− . . q (t; z) = 2n pf . . . 2n+1  2n 1  τ2n(t)  z − µ2n 1,2n+1   2n 1 2−n+1   1 ... z − 0 z   − − 2n+1 −   µ2n+1,0 ... µ2n+1,2n 1 z 0   −    2n 1/2 1 ∂ 1 2n+1 1/2 = z h− z + τ (t [z− ]) = z h− + .... 2n τ (t) ∂t 2n − 2n 2n  1  (3.2.15)

They are skew-orthogonal polynomials in z; i.e., q (t; z), q (t; z) = J . h i j it ij The hierarchy (3.2.11) already appears in the work of Kac-van de Leur [42] in the context of, what they call the DKP-hierarchy, and interesting further work has been done by van de Leur [71].

3.2.2 Sketch of Proof Skew-orthogonal polynomials and the Pfaff Lattice: The equalities (3.2.9) be- tween the Pfaffians and the matrix integrals are based on two identities [49], the first one due to de Bruyn,

1 n dyi det Fi(y1) Gi(y1) ... Fi(yn) Gi(yn) n! Rn 0 i 2n 1 1 ≤ ≤ − Z Y   1/2 = det (Gi(y)Fj(y) Fi(y)Gj(y))dy R − 0 i,j 2n 1 Z  ≤ ≤ − and (Mehta [50])

4 i i i i i i ∆n(x) = det x1 (x1)′ x2 (x2)′ ... xn (xn)′ 0 i 2n 1 . ≤ ≤ −
51 3, p.52 §

i tiz On the one hand, (see Mehta [49]), setting in the calculation below ρt(z)= ρ(z)e IE(z) P and x i i Fi(x) := y ρt(y)dy and Gi(x) := Fi′(x)= x ρt(x), Z−∞ i tiz one computes: (ρt(z) := ρ(z)e ) P 1 2n ∆2n(z) ρt(zi)dzi (2n)! R2n | | Z i=1 Y 2n = det zi ρ (z ) dz , j+1 t j+1 0 i,j 2n 1 i

= ρt(z2k)dz2k

= ρt(z2k)dz2k

i i j tix j tix ′ Fj(x)= x ρ(x)e and Gj(x) := Fj′(x)= x ρ(x)e , P P  

52 3, p.53 §

one computes

n ∞ i 1 4 2 2 tix (xi xj) ρ (xk)e i=1 k dxk n! Rn − 1 i,j n k=1 P Z ≤ ≤   Y n Y i 1 2 2 tix = ρ (xk)e k dxk n! Rn P Z k=1   Y i i i i i i det x1 (x1)′ x2 (x2)′ ... xn (xn)′ 0 i 2n 1 ≤ ≤ − n 1
= dyi det Fi(y1) Gi(y1) ... Fi(yn) Gi(yn) , n! Rn 0 i 2n 1 1 ≤ ≤ − Z Y   1/2 = det (Gi(y)Fj(y) Fi(y)Gj(y))dy R − 0 i,j 2n 1 Z  ≤ ≤ − ∞ i k ℓ 2tiy 2 = pf y ,y e 1 ρ (y)dy = τ2n(t), E{ } P 0 k,ℓ 2n 1 Z  ≤ ≤ − establishing the second equation (3.2.9). The skew-orthogonality of the polynomials qk(t; z) follows immediately from the skew-Borel decomposition of m : ∞ i j qk(t, y), qℓ(t, z) k,ℓ 0 = Q( y , z )i,j 0Q⊤ = Q m Q⊤ = J. h i ≥ h i ≥ ∞

with the qn’s admitting the representation (3.2.15) in terms of the moments. 1 1 1 2 Using L = QΛQ− , m = Q− JQ⊤− and J = I, one computes from the differ- ∞ ential equations (3.2.2) −

k k ∂m 0 = Q Λ m + m Λ⊤ ∞ Q⊤ ∞ ∞ − ∂t  k  k 1 1 k ∂Q 1 1 ∂Q⊤ = (QΛ Q− )J (JQ⊤− Λ⊤ Q⊤J)J + Q− J (JQ− ⊤ J)J − ∂tk − ∂tk

k ∂Q 1 k ∂Q 1 ⊤ = L + Q− J L + Q− J. ∂t − ∂t  k   k  Then computing the +, - and the diagonal part (in the sense of (3.2.4) and (3.2.5)) of the expression leads to commuting Hamiltonian differential equations for Q, and thus for L and q(t; z), confirming (3.2.13):

∂Q i ∂L i ∂q i = (L )kQ, = [(L )n, L], = (L )kq. (Pfaff lattice) (3.2.16) ∂ti − ∂ti ∂ti − The bilinear identities: For all n, m 0, the τ ’s satisfy the following bilinear ≥ 2n

53 3, p.54 §

identity

′ i 1 1 (ti t )z 2n 2m 2 dz τ2n(t [z− ])τ2m+2(t′ + [z− ])e − i z − − z= − P 2πi I ∞ (t′ t )z−i 2n 2m dz + τ (t + [z])τ (t′ [z])e i− i z − =0. 2n+2 2m − 2πi Iz=0 P (3.2.17)

The differential equation (3.2.2) on the moment matrix m admits the following solu- ∞1 1 tion, which upon using the Borel decomposition m = Q− J Q⊤− , leads to: ∞ 1 1 ∞ t Λk ∞ t Λ⊤k ∞ t Λk − ∞ t Λk ⊤− m (0) = e− 1 k m (t)e− 1 k = Q(t)e 1 k J Q(t)e 1 k , ∞ P ∞ P P P     (3.2.18) and so the right hand side of (3.2.11) is independent of t; say, equal to the same expression with t replaced by t′. Upon rearrangement, one finds

1 1 t Λk t′ Λk − t Λk ⊤− t′ Λk ⊤ Q(t)e k JQ(t′)e k = JQ(t)e k Q(t′)e k P P P P         and therefore24

1 1 ∞(t t′ )zk dz Q(t)χ(z) (JQ(t′))⊤− χ(z− ) e 1 k− k z= ⊗ P 2πiz I ∞ 1
1 ∞(t′ t )z−k dz = (JQ(t))⊤− χ(z) Q(t′)χ(z− ) e 1 k− k . (3.2.19) ⊗ 2πiz Iz=0 P 1 1
Setting t t′ = [z− ] + [z− ] into the exponential leads to − 1 2 1 1 ∞(t t′ )zk z − z − e 1 k− k = 1 1 − z − z P  1   2  ∞(t′ t )z−k 1 1 e 1 k− k = 1 1 − zz − zz P  1   2  and somewhat enlarging the integration circle about z = to include the points z and ∞ 1 z2, the integrand on the left hand side has poles at z = z1 and z2, whereas the integrand on the right hand side is holomorphic. Combining the identity obtained and the one, 1 with z , one finds a functional relation involving a function ϕ(t; z)=1+ O(z− ): 2 ր∞ 1 1 ϕ(t [z2− ]; z1) ϕ(t [z1− ]; z2) − = − , t C∞, z C. ϕ(t; z1) ϕ(t; z2) ∈ ∈ 24using Λχ(z)= zχ(z), Λ⊤χ(z)= z−1χ(z) and the following matrix identities (see [25]) dz dz U V = U χ(z) V ⊤χ(z−1) ,U V = U χ(z) V ⊤χ(z−1) . 1 1 1 ⊗ 1 2πiz 2 2 2 ⊗ 2 2πiz Iz=∞ Iz=0

54 3, p.55 §

Such an identity leads, by a standard argument (see e.g. the appendix in [8]) to the existence of a function τ(t) such that

1 τ(t [z− ]) ϕ(t; z)= − . τ(t)

This fact combined with the bilinear identity (3.2.19) leads to the bilinear identity (3.2.17). The Pfaff-KP-hierarchy: Shifting t t y, t′ t + y in (3.2.17), evaluating the 7→ − 7→ residue and Taylor expanding in yk leads to:

∞ i 1 2yiz 1 1 2n 2m 2 e− 1 τ2n(t y [z− ])τ2m+2(t + y + [z− ])z − − dz 2πi z= P − − I ∞ 1 ∞ 2y z−i 2n 2m + e 1 i τ (t y + [z])τ (t + y [z])z − dz 2πi 2n+2 − 2m − Iz=0 P 1 ∞ ∂ ∞ j yi ∂t k 2n 2m 2 = z sj( 2y)e − i z− sk( ∂˜)τ2n τ2m+2z − − dz 2πi − P − ◦ z= j=0 I ∞ X Xk=0 1 ∞ ∂ ∞ j yi ∂t k 2n 2m + z− sj(2y)e − i z sk(∂˜)τ2n+2 τ2mz − dz 2πi P ◦ z=0 j=0 I X Xk=0

∂ yi ∂t = sj( 2y)e − i sk( ∂˜)τ2n τ2m+2 − P − ◦ j k= 2n+2m+1 − − X ∂ yi ∂t + sj(2y)e − i sk(∂˜)τ2n+2 τ2m P ◦ k j= 2n+2m 1 − −X − 1 ∂ ∂ = ... + yk sk+1(∂˜) τ2n τ2n + sk 3(∂˜)τ2n+2 τ2n 2 + ..., 2 ∂t ∂t − ◦ − ◦ −  1 k   establishing the Pfaff-KP hierarchy (3.2.11), different from the usual KP hierarchy, because of the presence of a right hand side. Remark: L admits the following representation in terms of τ, much in the style of (3.1.7),

Lˆ00 Lˆ01 0 0 Lˆ Lˆ Lˆ 0  10 11 12  1/2 ˆ ˆ ˆ 1/2 L = h− L21 L22 L23 h ,  ∗ ˆ ˆ   L32 L33 ...   ∗ ∗ .   .     

55 3, p.56 §

with the 2 2 entries Lˆij and h, being a zero matrix, except for 2 2 matrices along the diagonal:× × h = diag(h0I2, h2I2, h4I2,... ), h2n = τ2n+2/τ2n. For example (. = ∂ ) ∂t1

. (log τ2n) 1 ˆ − ˜ ˜ Lnn := s2(∂)τ2n s2( ∂)τ2n+2 . − (log τ2n+2) − τ2n − τ2n+2 ! 0 0 (log τ ).. Lˆ := Lˆ := 2n+2 . n,n+1 1 0 n+1,n ∗    ∗ ∗ 

3.3 2d-Toda lattice and coupled Hermitian matrix integrals 3.3.1 2d-Toda lattice, factorization of moment matrices and bi-orthogonal polynomials Consider the inner-product,

∞ i i (tiy siz )+cyz f,g t,s = f(y)g(z)e 1 − dy dz, (3.3.1) h i E R2 P Z Z ⊂ r s 2 on a subset E = E1 E2 := i=1[c2i 1,c2i] i=1[˜c2i 1, c˜2i] F1 F2 R . Define the × ∪ − × ∪ − ⊂ × ⊂ customary moment matrix, depending on t =(t1, t2,... ), s =(s1,s2,... ),

i j mn(t, s)=(µij(t, s))0 i,j n 1 =( y , z t,s)0 i,j n 1 ≤ ≤ − h i ≤ ≤ − and its factorization in lower- times upper-triangular matrices

1 m (t, s)= S1− (t, s)S2(t, s). (3.3.2) ∞ Then m evolves in t, s according to the equations ∞

∂µij ∂µij ∂m k ∂m k = µi+k,j, = µi,j+k, i.e., ∞ =Λ m , ∞ = m Λ⊤ . (3.3.3) ∂tk ∂sk − ∂tk ∞ ∂sk − ∞ dM in the integral (3.3.4) denotes properly normalized Haar measure on . Hn

Theorem 3.4 (Adler-van Moerbeke [4, 3]) The integrals In(t, s; c, c˜), with I0 =1,

n 1 1 ∞ i i (tix siy )+cxkyk τn = det mn = In = ∆n(x)∆n(y) e 1 k− k dxkdyk n! n! En P ZZ Yk=1 ∞ i i c Tr(M1M2) Tr (tiM siM ) = e e 1 1− 2 dM1dM2, (3.3.4) 2 (E) P ZZHn satisfy:

56 3, p.57 §

(i) Virasoro constraints25 (2.2.8) for k 1, ≥ − r k+1 ∂ (2) E ˜ ˜ E E ci + Jk,n τn + c sk+n(∂t)sn( ∂s)τ1 τn 1 = 0 − ∂c − ◦ − i=1 i ! Xs k+1 ∂ ˜(2) E ˜ ˜ E E c˜i + Jk,n τn + c sn(∂t)sk+n( ∂s)τ1 τn 1 = 0, − ∂c˜ − ◦ − i=1 i ! X (3.3.5)

with 1 J (2) = (J (2) + (2n + k + 1)J (1) + n(n + 1)J (0)), k,n 2 k k k 1 J˜(2) = (J˜(2) + (2n + k + 1)J˜(1) + n(n + 1)J (0)). k,n 2 k k k

(ii) A Wronskian identity 26:

∂2 log τ ∂2 log τ ∂2 log τ ∂2 log τ n , n + n , n =0. (3.3.6) ∂t1∂s2 ∂t1∂s1 ∂s1∂t2 ∂t1∂s1  t1  s1

1 (iii) The 2d-Toda lattice: Given the factorization (3.3.2), the matrices L1 := S1ΛS1− and L2 := 1 τn+1 S Λ⊤S− , with h = , have the form (i.e., to be read as follows: the (k ℓ)th 2 2 n τn k ℓ − subdiagonal is given by the diagonal matrix in front of Λ − )

∞ ˜ k sℓ(∂t)τn+k ℓ+1 τn k ℓ L1 = diag − ◦ Λ − τn+k ℓ+1τn ℓ=0 − !n Z X ∈ ∞ ˜ k 1 sℓ( ∂s)τn+k ℓ+1 τn k ℓ hL2⊤ h− = diag − − ◦ Λ − , (3.3.7) τn+k ℓ+1τn ℓ=0 − !n Z X ∈ and satisfy the 2d-Toda Lattice27

∂Li n ∂Li n = [(L1 )+ , Li] and = [(L2 ) , Li] , i =1, 2, (3.3.8) ∂tn ∂sn −

25 (i) For the Hirota symbol, see footnote 14. The Jk ’s are as in remark 1 at the end of Theorem 2.1, (i) (i) for β = 1 and J˜ = J →− , with k k |t s 2 (1) ∂ (2) ∂ ∂ J = + ( k)t− , J = +2 it + it jt . k ∂t − k k ∂t ∂t i ∂t i j k i j − j − − i+Xj=k iX+j=k iXj=k

26 ∂f ∂g in terms of the Wronskian f,g t = ∂t g f ∂t . 27 { } − P+ and P− denote the upper (including diagonal) and strictly lower triangular parts of the matrix P , respectively.

57 3, p.58 §

(iv) Bi-orthogonal polynomials: The expressions

1 (1) n τn(t [y− ],s) pn (t, s; y) := (S1(t, s)χ)y))n = y − τn(t, s) 1 (2) 1 n τn(t, s + [z− ]) pn (t, s; z) := hS2⊤− (t, s)χ(z) n = z (3.3.9) τn(t, s)
form a system of monic bi-orthogonal polynomials in z:

(1) (2) τn+1 pn (t, s; y),pm (t, s; z) t,s = δn,mhn with hn = , (3.3.10) h i τn

which also are eigenvectors of L1 and L2:

(1) (1) (2) (2) zpn (t, s; z)= L1(t, s)pn (t, s; z) and zpn (t, s; z)= L2⊤(t, s)pn (t, s; z). (3.3.11)

Remark: Notice that every statement can be dualized, upon using the duality t s, 1 ←→ − L hL⊤h− . 1 ←→ 2 3.3.2 Sketch of proof Identity (3.3.4) follows from the fact that the product of the two Vandermonde appear- ing in the integral (3.3.4) can be expressed as sum of determinants:

ℓ 1 k 1 ∆n(u)∆n(v)= det uσ−(k)vσ(−k) , (3.3.12) 1 ℓ,k n σ Sn ≤ ≤ X∈   together with the Harish-Chandra, Itzykson and Zuber formula [34, 38]

n(n−1) cxiyj c Tr xUyU¯ ⊤ (2π) 2 det(e )1 i,j n dU e = ≤ ≤ . (3.3.13) n! ∆ (x)∆ (y) ZU(n) n n Moreover the τ ’s satisfy the following bilinear identities, for all integer m, n 0 and n ≥ t, s C∞: ∈

∞ ′ i 1 1 (ti t )z n m 1 τn(t [z− ],s)τm+1(t′ + [z− ],s′)e 1 − i z − − dz z= − P I ∞

∞(s s′ )z−i n m 1 = τ (t, s [z])τ (t′,s′ + [z])e 1 i− i z − − dz. (3.3.14) n+1 − m Iz=0 P Again, the bi-orthogonal nature (3.3.10) of the polynomials (3.3.9) is tantamount to 1 the Borel decomposition, written in the form S1m (hS2⊤− )⊤ = h. These polynomials ∞

58 3, p.59 §

satisfy the eigenvalue problem (3.3.11) and evolve in t, s according to the differential equations (1) (1) ∂p n (1) ∂p n (1) = (L1 ) p = (L2 ) p ∂tn − − ∂sn − − (2) (2) ∂p 1 n (2) ∂p 1 n (2) = (h− L1h)⊤ p = (h− L2h)⊤ p . (3.3.15) ∂tn − − ∂sn − From the representation (3.3.7)
and from the bilinear identity
(3.3.14), it follows that 2 pk 1(∂˜t)τn+2 τn ∂ − 2 ◦ = log τn+1, (3.3.16) τn+1 −∂s1∂tk and so, for k = 1, 2 τnτn+2 ∂ 2 = log τn+1. (3.3.17) τn+1 −∂s1∂t1 Thus, using (3.3.7), (3.3.16) and (3.3.17), we have

2 ˜ ∂ log τn+1 k pk 1(∂t)τn+2 τn ∂s1∂tk − L1 n,n+1 = ◦ = ∂2 log τ τn+2τn n+1 ∂s1∂t1
2 ˜ ∂ log τn+1 k 1 pk 1( ∂s)τn+2 τn ∂t1∂sk ⊤ − − hL2 h n,n+1 = − ◦ = ∂2 log τ . (3.3.18) τn+2τn n+1 ∂s ∂t
1 1 Combining (3.3.18) with (3.3.17) for k = 2 yields

∂2 log τn+1 2 ∂s1∂t2 ∂ τn+2 L1 n,n+1 = ∂2 = log (3.3.19) log τn+1 ∂t1 − τn ∂s1∂t1  
∂ τ 2 ∂2 = log n+1 log τ . ∂t τ ∂s ∂t n+1 1  n  1 1 !

Then, subtracting ∂/∂s1 of (3.3.19) from ∂/∂t1 of the dual of the same equation (see remark at the end of Theorem 3.4) leads to (3.3.6).

3.4 The Toeplitz Lattice and Unitary matrix integrals 3.4.1 Toeplitz lattice, factorization of moment matrices and bi-orthogonal polynomials Consider the inner-product

∞ i −i dz 1 (tiz siz ) f(z),g(z) t,s := f(z)g(z− )e 1 − , t,s C∞, (3.4.1) h i 1 2πiz ∈ IS P

59 3, p.60 §

where the integral is taken over the unit circle S1 C around the origin. It has the property ⊂

k k z f,g = f, z− g . (3.4.2) h it,s h it,s The t, s dependent semi-infinite moment matrix m (t, s), where28 ∞

ρ(z)dz ∞ i −i k ℓ k ℓ 1 (tiz siz ) mn(t, s) := z , z t,s 0 k,ℓ n 1 = z − e − h i ≤ ≤ − S1 2πiz P 0 k,ℓ n 1 I  ≤ ≤ −
= Toeplitz matrix (3.4.3)

satisfies the same differential equations, as in (3.3.3):

∂m n ∂m n ∞ =Λ m and ∞ = m Λ⊤ . (2-Toda Lattice) (3.4.4) ∂tn ∞ ∂sn − ∞ As before, define τn(t, s) := det mn(t, s). 1 Also, consider the factorization m (t, s) = S1− (t, s)S2(t, s), as in (3.3.2), from which 1 ∞ 1 one defines L1 := S1ΛS1− and L2 := S2Λ⊤S2− and the bi-orthogonal polynomials (k) pi (t, s; z) for k = 1, 2. Since m satisfies the same equations (3.3.3), the matrices ∞ L1 and L2 satisfy the 2-Toda lattice equations; the Toeplitz nature of m implies a hi ∞ peculiar “rank 2”-structure, with =1 xiyi and x0 = y0 = 1 : hi−1 − x y 1 x y 0 0 − 1 0 − 1 1 x2y0 x2y1 1 x2y2 0 1  − − −  h− L1h = x3y0 x3y1 x3y2 1 x3y3  −x y −x y −x y −x y   4 0 4 1 4 2 4 3   − − − − .   ..      and

x0y1 x0y2 x0y3 x0y4 1− x y −x y −x y −x y  1 1 1 2 1 3 1 4  0− 1− x y −x y −x y L2 = − 2 2 − 2 3 − 2 4 . (3.4.5)  0 0 1 x y x y   3 3 3 4   − − .   ..      Some of the ideas in the next theorem are inspired by the work of Hisakado [37]. 28 A matrix is Toeplitz, when its (i, j)th entry depends on i j. −

60 3, p.61 §

Theorem 3.5 (Adler-van Moerbeke [7]) The integrals In(t, s), with I0 =1,

n ∞ i −i 1 1 2 (tiz siz ) dzk τn(t, s) = det mn = In : = ∆n(z) e 1 k− k n! n! (S1)n | | P 2πizk Z Yk=1   ∞ Tr(t M i s M¯ i) = e 1 i − i dM ZU(n) P = s (t)s ( s), (3.4.6) λ λ − Young diagrams λ λˆ1 n { X | ≤ } satisfy:

(i) a SL(2,Z)-algebra of three Virasoro constraints (2.3.2):

J(2) J(2) J(1) J(1) k,n(t, n) k,n( s, n) k θ k,n(t, n)+(1 θ) k,n( s, n) In(t, s)=0, − − − − − − −   (3.4.7) k = 1, θ =0 − for k =0, θ arbitrary   k =1, θ =1

(ii) 2d-Toda identities: The matrices L1 and L2, defined above, satisfy the 2-Toda lat- tice equations (3.3.8); in particular:

2 ∂ τn 1τn+1 − log τn = 2 ∂s1∂t1 − τn and

2 2 3 ∂ ∂ τn ∂ ∂ log τn = 2 log . log τn 2 log τn, (3.4.8) ∂s2∂t1 − ∂s1 τn 1 ∂s1∂t1 − ∂s1∂t1 − the first being equivalent to the discrete sinh-Gordon equation

2 ∂ qn qn qn−1 qn+1 qn τn+1 = e − e − , where qn = log . ∂t1∂s1 − τn

(iii) The Toeplitz lattice : The 2-Toda lattice solution is a very special one, namely the matrices L1 and L2 have a ”rank 2” structure, given by (3.4.5), whose xn’s and

61 3, p.62 §

29 yn’s equal :

xn(t, s)

1 1 2 1 3 ∞ Tr(t M i s M¯ i) = s ( Tr M, Tr M , Tr M , ...)e 1 i − i dM τ n − −2 −3 n ZU(n) P ∂ 1 ∂ 1 ∂ sn( , , , ...)τn(t, s) ∂t1 2 ∂t2 3 ∂t3 (1) = − − − = pn (t, s;0) τn(t, s)

yn(t, s)

1 1 2 1 3 ∞ Tr(t M i s M¯ i) = s ( Tr M,¯ Tr M¯ , Tr M¯ , ...)e 1 i − i dM τ n − −2 −3 n ZU(n) P ∂ 1 ∂ 1 ∂ sn( , , , ...)τn(t, s) ∂t1 2 ∂t2 3 ∂t3 (2) = = pn (t, s;0), (3.4.9) τn(t, s) and satisfy the following integrable Hamiltonian system

(1) (1) ∂xn ∂Hi ∂yn ∂Hi = (1 xnyn) = (1 xnyn) ∂ti − ∂yn ∂ti − − ∂xn (2) (2) ∂xn ∂Hi ∂yn ∂Hi = (1 xnyn) = (1 xnyn) , (3.4.10) ∂si − ∂yn ∂si − − ∂xn (Toeplitz lattice)

with initial condition x (0, 0) = y (0, 0) = 0 for n 1 and boundary condition n n ≥ x0(t, s)= y0(t, s)=1. The traces 1 H(k) = T r Li , i =1, 2, 3, ..., k =1, 2. i − i k

of the matrices Li in (3.4.5) are integrals in involution with regard to the sym- 1 plectic structure ω := ∞(1 xkyk)− dxk dyk. The Toeplitz nature of m leads 0 ∞ to identities between τ’s, the− simplest being∧ (due to Hisakado [37]) : P ∂2 ∂2 ∂ τ ∂ τ 1+ log τ 1+ log τ = log n+1 log n+1 . ∂s ∂t n+1 ∂s ∂t n −∂t τ ∂s τ  1 1   1 1  1 n 1 n (3.4.11)

Remark: The first equation in the hierarchy above reads:

∂xn ∂yn = xn+1(1 xnyn) = yn 1(1 xnyn) ∂t1 − ∂t1 − − − ∂xn ∂yn = xn 1(1 xnyn) = yn+1(1 xnyn). ∂s1 − − ∂s1 − −

29 Remember s(t1,t2, ...) are elementary Schur polynomials.

62 3, p.63 §

3.4.2 Sketch of Proof The identity (3.4.6) between the determinant and the moment matrix uses again the Vandermonde identity (3.3.12),

∞ Tr(t M i s M¯ i) e 1 i − i dM ZU(n) P n ∞ i −i 2 (tiz siz ) dzk = ∆n(z) e 1 k− k 1 n | | 2πizk Z(S ) k=1  P  Y n ∞ i −i (tiz siz ) dzk = ∆n(z)∆n(¯z) e 1 k− k 1 n 2πizk Z(S ) k=1  P  Y n ∞ i −i dzk ℓ 1 m 1 1 (tizk sizk ) = det zσ−(m)z¯σ(−m) e − (S1)n 1 ℓ,m n P 2πizk σ Sn ≤ ≤ k=1   Z X∈   Y ∞ i −i dzk ℓ 1 m 1 1 (tizk sizk ) = det zk− z¯k − e − S1 P 2πizk 1 ℓ,m n σ Sn   X∈ I ≤ ≤ ∞ i −i ℓ m (tiz siz ) dz = n! det z − e 1 − = n! det mn(t, s)= n!τn(t) S1 P 2πiz 1 ℓ,m n I  ≤ ≤

k k (1) (1) Using z ⊤ = z− (see (3.4.2)), one shows that the polynomials pn+1(z) zpn (z) (1) n (2) 1 0 1 n − and pn+1(0)z pn (z− ) are perpendicular to the monomials z , z , ..., z and that they have the same z0-term; one makes a similar argument, by dualizing 1 2. Therefore, ↔ we have the Hisakado identities between the following polynomials:

(1) (1) (1) n (2) 1 p (z) zp (z) = p (0)z p (z− ) n+1 − n n+1 n (2) (2) (2) n (1) 1 p (z) zp (z) = p (0)z p (z− ). (3.4.12) n+1 − n n+1 n (1) (2) The rank 2 structure (3.4.5) of L1 and L2, with xn = pn (t, s; 0) and yn = pn (t, s; 0), (1) (1) is obtained by taking the inner-product of pn+1(z) zpn (z) with itself, for different n (1) (1) − and m, and using the fact that zpn (z)= L1pn (z). To check the first equation in the hierarchy (see remark at the end of theorem 3.5),

63 4, p.64 §

consider, from (3.4.9),

∂x ∂p(1)(t, s; z) n = n ∂t1 ∂t1 z=0 (1) = (L1) p , using (3.3.15), − − n z=0 n 1 (1) (2)
− p (t, s;0)p (t, s;0) = h p(1) (t, s;0) i i , using (3.4.5), n n+1 h i=0 i n 1 X − x y = h x i i n n+1 h i=0 i Xn 1 − 1 1 h = h x , using i =1 x y , n n+1 h − h h − i i i=0 i i 1 i 1 X  −  − hn = xn+1 hn 1 − = x (1 x y ), n+1 − n n and similarly for the other coordinates. From (3.3.7) and (3.4.5), upon making the 1 products of the corresponding diagonal entries of L1 and hL2⊤h− , one finds (3.4.11):

∂ τn+1 ∂ τn+1 log log = xn+1ynxnyn+1 = xnynxn+1yn+1 ∂t1 τn ∂s1 τn − − h h = 1 n 1 n+1 . − − hn 1 − hn  −   

4 Ensembles of finite random matrices

4.1 PDE’s defined by the probabilities in Hermitian, symmet- ric and symplectic random ensembles

2r V (z) As used earlier, the disjoint union E = 1 [c2i 1,c2i] R, and the weight ρ(z)= e− , ∪ − ⊂ with ρ′/ρ = V ′ = g/f define an algebra of differential operators, (k Z) − ∈ 2r ∂ = ckf(c ) . Bk i i ∂c 1 i X

64 4, p.65 §

The aim of this section is to find PDE’s for the following probabilities in terms of the boundary points ci of E (see (1.1.9), (1.1.11) and (1.1.18)), i.e.

Pn(E): = Pn( all spectral points of M E) tr V (M) ∈ e− dM n(E), n(E) or n(E) = H S T (4.1.1) e tr V (M)dM R n(R), n(R) or n(R) − H S T β n V (zk) REn ∆n(z) k=1 e− dzk = | | n β =2, 1, 4 respectively, ∆ (z) β e V (zk)dz RRn | n | Qk=1 − k involving the classicalR weights below.Q In anticipation, the equations obtained in Theo- rems 4.1, 4.2 and 4.3 are closely related to three of the six Painlev´edifferential equations:

weight ρ(z) Painlev´e bz2 Gauss e− IV a bz Laguerre z e− V Jacobi (1 z)a(1 + z)b VI − For β = 2, the probabilities satisfy partial differential equations in the boundary points of E, whereas in the case β =1, 4, the equations are inductive. Namely, for β = 1 (resp. β = 4) , the probabilities Pn+2 (resp. Pn+1) are given in tems of Pn 2 (resp. Pn 1) and − − a differential operator acting on Pn. The weights above involve the parameters β, a, b and 1/2 1/2 2 β β − 0 for β =2 δβ := 2 = 1,4 2 − 2 1 for β =1, 4.     !  As a consequence of the duality (2.1.12) between β-Virasoro generators under the map β 4/β, and the equations (2.1.7), the PDE’s obtained have a remarkable property: 7→ the coefficients Q and Qi of the PDE’s are functions in the variables n, β, a, b, having the invariance property under the map a b n 2n, a , b ; → − → −2 → −2 to be precise,

a b Q ( 2n, β, , ) = Q (n, β, a, b) . (4.1.2) i − −2 −2 i |β=4 β=1

The results in this section are mainly due to Adler-Shiota-van Moerbeke [11] for β = 2 and Adler-van Moerbeke [6] for β = 1, 4. For more detailed references, see the end of section 4.2.

65 4, p.66 §

4.1.1 Gaussian Hermitian, symmetric and symplectic ensembles

bz2 Given the disjoint union E and the weight e− , the differential operators k take on the form B 2r ∂ = ck+1 . Bk i ∂c 1 i X Define the invariant polynomials 2 2 Q = 12b2n n +1 and Q = 4(1+ δβ )b 2n + δβ (1 ) . − β 2 1,4 1,4 − β    

Theorem 4.1 The following probabilities (β =2, 1, 4)

n 2 β bzk n ∆n(z) e− dzk P (E)= E | | k=1 , (4.1.3) n n bz2 ∆ (z) β e k dz RRn | n | Qk=1 − k

satisfy the PDE’s (F := Fn = log PRn): Q

Pn 2 Pn+ 2 2 when n even and β =1 δβ Q − 1 1 1 with index 1,4 P 2 − 1 when n arbitrary and β =4 n !  2 4 2 2 β b 2 = 1 +(Q2 +6 1F ) 1 + 4(2 δ1,4) (3 0 4 1 1 +6 0) F. (4.1.4) B− B− B− − β B − B− B B   4.1.2 Laguerre Hermitian, symmetric and symplectic ensembles

+ a bz Given the disjoint union E R and the weight z e− , the take on the form ⊂ Bk 2r ∂ = ck+2 . Bk i ∂c 1 i X Define the polynomials, also respecting the duality (4.1.2), 3 n(n 1)(n +2a)(n +2a + 1), for β =1 4 − Q =   3  n(2n + 1)(2n + a)(2n + a 1), for β =4 2 −  a2 β Q = 3βn2 +6an + 4(1 )a +3 δβ + (1 a2)(1 δβ ) 2 − β − 2 1,4 − − 1,4   β a Q = βn2 +2an + (1 )a , Q = b(2 δβ )(n + ). 1 − 2 0 − 1,4 β  

66 4, p.67 §

Theorem 4.2 The following probabilities

β n a bzk En ∆n(z) k=1 zk e− dzk Pn(E)= | | n (4.1.5) β a bzk Rn ∆n(z) k=1 zk e− dzk R + | | Q 30 satisfy the PDE : (F := Fn = logR Pn) Q Pn 2 Pn+ 2 β − 1 1 δ1,4Q 2 1 Pn − !

4 β 3 2 β 2 β = 1 2(δ1,4 +1) 1 +(Q2 +6 1F 4(δ1,4 +1) 1F ) 1 3δ1,4(Q1 1F ) 1 − − − B− − 2 B− B− − B B− − − B B  b β 2 + (2 δ1,4)(3 0 4 1 1 2 1)+ Q0(2 0 1 0) F. (4.1.6) β − B − B B− − B B B− − B 

4.1.3 Jacobi Hermitian, symmetric and symplectic ensembles In terms of E [ 1, +1] and the Jacobi weight (1 z)a(1+z)b, the differential operators ⊂ − − k take on the form B 2r ∂ = ck+1(1 c2) . Bk i − i ∂c 1 i X Introduce the following variables, which themselves have the invariance property (4.1.2) (b = a b, b = a + b): 0 − 1 4 4 r = (b2 +(b +2 β)2) s = b (b +2 β) β 0 1 − β 0 1 − 4 q = (βn + b +2 β)(βn + b ), n β 1 − 1 and the following invariant polynomials in q,r,s: 3 Q = (s2 qr + q2)2 4(rs2 4qs2 4s2 + q2r) 16 − − − − Q = 3s2 3qr 6r +2q2 + 23q + 24
1 − − Q = 3qs2 +9s2 4q2 r +2qr +4q3 + 10q2 2 − Q = 3qs2 +6s2 3q2r + q3 +4q2 3 − Q = 9s2 3qr 6 r + q2 + 22q +24= Q + (6s2 q2 q). 4 − − 1 − − (4.1.7)

Theorem 4.3 The following probabilities ∆ (z) β n (1 z )a(1 + z )bdz En | n | k=1 − k k k Pn(E)= β n a b (4.1.8) [ 1,1]n ∆n(z) k=1(1 zk) (1 + zk) dzk R− | |Q − satisfy the PDE (F = Fn =R log Pn): Q 30with the same convention on the indices n 2 and n 1, as in (4.1.4) ± ± 67 4, p.68 § for β =2:

4 2 2 2 2 1 +(q r + 4) 1 (4 1F s) 1 +3q 0 2q 0 +8 0 1 B− − B− − B− − B− B − B B B−  4(q 1) 1 1 + (4 1F s) 1 + 2(4 1F s) 0 1 +2q 2 F − − B B− B− − B B− − B B− B 2 2  +4 1F 2 0F +3 1F = 0 (4.1.9) B− B B− for β =1, 4:

P 2 P 2 n+ 1 n− 1 Q 2 1 Pn − !

4 3 = (q + 1) 4q + 12(4 − F s) + 2 (q + 12)(4 − F s) − B−1 B 1 − B−1 B 1 − B0B 1 2 2  2 2 +3q 4 (q 4) q − + q(4 − F s) + 20q +2q F B0 − − B1B 1 B 1 − B1 B0B−1 B2 2 4 3  2 + Q sQ − + Q F + 48( − F ) 48s( − F ) +2Q ( − F ) 2B−1 − 1B 1 3B0 B 1 − B 1 4 B 1 +12 q2( F )2 + 16 q (2 q 1) ( 2 F )( F )+24(q 1) q( 2 F )2 B0 − B−1 B0 − B−1 2 +24 2 − F s (q + 2) F + (q + 3) F − F. (4.1.10) B 1 − B0 B−1 B 1    The Proof of these three theorems will be sketched in subsections 4.3, 4.4 and 4.5.

4.2 ODE’s, when E has one boundary point Assume the set E consists of one boundary point c = x, besides the boundary of the full range; thus, setting respectively E = [ , x], E = [0, x], E = [ 1, x] in the PDE’s −∞ − (4.1.4), (4.1.6) and (4.1.9), (4.1.10), leads to the equations in x below. Notice that, for β = 2, the equations obtained are ODE’s and, for β = 1, 4, these equations express Pn+2 in terms of Pn 2 and a differential operator acting on Pn: − (1) Gauss ensemble (β =2, 1, 4) : f (x)= d log P (max λ x) satisfies n dx n i i ≤

Pn 2 Pn+ 2 β − 1 1 δ1,4Q 2 1 (4.2.1) Pn − ! 2 2 2 2 b x β b x β = f ′′′ +6f ′ + 4 (δ 2) + Q f ′ 4 (δ 2)f . n n β 1,4 − 2 n − β 1,4 − n   (2) Laguerre ensemble (β = 2, 1, 4): f (x) = x d log P (max λ x) (with all eigen- n dx n i i ≤

68 4, p.69 §

values λ 0) satisfies: i ≥ 2 2 Pn 2 Pn+ 2 b x δβ Q − 1 1 1 3δβ f (δβ 2) Q x 3δβ Q f 1,4 P 2 − − 1,4 n − β 1,4 − − 0 − 1,4 1 n n !   3 β 2 2 2 = x f ′′′ (2δ 1)x f ′′ +6x f ′ n − 1,4 − n n 2 2 β b x β β x 4(δ + 1)f (δ 2) 2Q x Q +2δ +1 f ′ . − 1,4 n − β 1,4 − − 0 − 2 1,4 n   (4.2.2)

2 d (3) Jacobi ensemble: f := fn(x) =(1 x ) dx log Pn(maxi λi x) (with all eigenvalues 1 λ 1) satisfies: − ≤ − ≤ i ≤ for β =2: • 2 2 2 2 2 2(x 1) f ′′′ + 4(x 1) xf ′′ 3f ′ + 16xf q(x 1) 2sx r f ′ − − − − − − − f (4f qx s)=0, (4.2.3) − − −

for β =1, 4: • Pn+ 2 Pn 2 1 − 1 Q 2 1 Pn − !

= 4(q + 1)(x2 1)2 q(x2 1)f ′′′ + (12f qx 3s)f ′′ +6q(q 1)f ′2 − − − − − −   (x2 1)f ′ 24f(q + 3)(2f s)+8fq(5q 1)x q(q + 1)(qx2 +2sx +8)+ Q − − − − − 2   +f 48f 3 + 48f 2(qx +2x s)+2f 8q2x2 +2qx2 12qsx 24sx + Q − − − 4  q(q + 1)x(3qx2 + sx 2qx 3q)+ Q x Q s .
(4.2.4) − − − 3 − 1 

P 2 P 2 n+ 1 n− 1 For β = 2, the term containing the ratio 2 1 on the left hand side of (4.2.1), Pn − (4.2.2) and (4.2.4) vanishes and one thus obtains the ODE’s:

Gauss: f (x) := d log P (max λ x) satisfies: • n dx n i i ≤ 2 2 2 f ′′′ +6 f ′ +4b(2n bx )f ′ +4b x f =0 − Laguerre: f (x) := x d log P (max λ x) satisfies • n dx n i i ≤ 2 2 2 x f ′′′ + xf ′′ +6xf ′ 4ff ′ ((a bx) 4nbx)f ′ b(2n + a bx)f =0. − − − − − −

69 4, p.70 §

Jacobi: f (x)=(1 x2) d log P (max λ x) satisfies: • n − dx n i i ≤

2 2 2 2 2 2(x 1) f ′′′ + 4(x 1) xf ′′ 3f ′ + 16xf q(x 1) 2sx r f ′ − − − − − − − f (4f qx s)=0.
− − −
Each of these three equations is of the Chazy form (see section 9)

P ′ 6 2 4P ′ P ′′ 2 4Q 2Q′ 2R f ′′′ + f ′′ + f ′ ff ′ + f + f ′ f + =0, (4.2.5) P P − P 2 P 2 P 2 − P 2 P 2 with c = 0 and P, Q, R having the form:

Gauss P (x)=1 4Q(x)= 4b2x2 +8bn R =0 Laguerre P (x)= x 4Q(x)= −(bx a)2 +4bnx R =0 − − Jacobi P (x)=1 x2 4Q(x)= 1 (q(x2 1)+2sx + r) R =0 − − 2 − Cosgrove shows such a third order equation (4.2.5) in f(x), with P (x), Q(x), R(x) of respective degrees 3, 2, 1, has a first integral (9.0.2), which is second order in f and quadratic in f ′′, with an integration constant c. Equation (9.0.2) is a master Painlev´e equation, containing the 6 Painlev´eequations. If f(x) satisfies the equations above, then the new (renormalized) function g(z), defined below,

1/2 1/2 2 Gauss g(z)= b− f(zb− )+ 3 nz b a2 Laguerre g(z)= f(z)+ 4 (2n + a)z + 4 1 q q+s Jacobi g(z) := f(x) x=2z 1 z + − 2 | − − 8 16 satisfies the canonical equations, which then can be transformed into the standard Painlev´eequations; these canonical equations are respectively:

2 3 2 g′′ = 4g′ + 4(zg′ g) + A g′ + A (Painlev´eIV) • − − 1 2 2 2 (zg′′) =(zg′ g) 4g′ + A (zg′ g)+ A + A g′ + A • − − 1 − 2 3 4   (Painlev´eV)

2 2 2 (z(z 1)g′′) =(zg′ g) 4g′ 4g′(zg′ g)+ A + A g′ + A g′ + A • − − − − 2 1 3 4 with respective coefficients   (Painlev´eVI)

A =3 4n 2 , A = 4n 3 • 1 3 2 − 3 2
2 a 2
a2 2 a (ab)2 A1 = b , A2 = b ((n + 2 ) + 2 ), A3 = a b(n + 2 ), A4 = 2 • − a 2 a2 .((n + 2 ) + 8 )

70 4, p.71 §

2q+r qs (q s)2+2qr q 2 A = , A = , A = − , A = (2s + qr). • 1 8 2 16 3 64 4 512 For β = 1 and 4, the inductive partial differential equations (4.1.4), (4.1.6), (4.1.10), and the derived differential equations (4.2.1), (4.2.2) and (4.2.4) are due to Adler-van Moerbeke [6]. For β = 2 and for general E, they were first computed by Adler-Shiota- van Moerbeke [11], using the method of the present paper. For β = 2 and for E having one boundary point, the equations obtained here coincide with the ones first obtained by Tracy-Widom in [64], who recognized them to be Painlev´eIV and V for the Gaussian and Laguerre distribution respectively. In his Louvain doctoral dissertation, J.P. Semengue, together with L. Haine [32], were lead to Painlev´eVI for the Jacobi ensemble, for β = 2 and E having one boundary point, upon subtracting the Tracy- Widom differential equation ([64]) from the one computed with the Adler-Shiota-van Moerbeke method ([11]). Cosgrove’s ([23]) and Cosgrove-Scoufis’s classification ([24], (A.3),) leads directly to these results.

4.3 Proof of Theorems 4.1, 4.2 and 4.3 4.3.1 Gaussian and Laguerre ensembles The three first Virasoro equations, as in (2.1.29) and (2.1.32), are differential equations, 2r involving partials in t C∞ and partials , , in c = (c , ..., c ) R , for ∈ D1 D2 D3 1 2r ∈ F := Fn(t, c) = log In; they have the general form: ∂F kF = + γkjVj(F )+ γk + δkt1, k =1, 2, 3, (4.3.1) D ∂tk 1 j

with first Vj(F )’s given by:

∂F β ∂2F ∂F 2 V (F )= it + δ + , 1 j 2. (4.3.2) j i ∂t 2 2,j ∂t2 ∂t − ≤ ≤ i,i+j 1 i+j 1  1  ! X≥

In (4.3.1) and (4.3.2), β > 0,γkj,γk, δk are arbitrary parameters; also δ2j = 0 for j =2 and = 1 for j = 2. The claim is that the equations (4.3.1) enable one to express6 all partial derivatives,

∂i1+...+ik F (t, c) , along := all ti =0, c =(c1, ..., c2r) arbitrary , (4.3.3) ∂ti1 ...∂tik L { } 1 k L uniquely in terms of polynomials in ... F (0,c). Dj1 Djr

Indeed, the method consists of expressing ∂F/∂tk t=0 in terms of kf t=0, using (4.3.1). Second derivatives are obtained by acting on F with , by notingD that commutes Dk Dℓ Dℓ

71 4, p.72 §

with all t-derivatives, by using the equation for F , and by setting in the end t = 0: Dℓ ∂F F = + γ (V (F )) DℓDk Dℓ ∂t kj Dℓ j k − ≤ 1Xj

2 ∂ F 2 2 = 1 γ10 1 F + γ10γ1 δ1 ∂t1 D − D − L
∂4F 4 3 2 2 3 2 4 = 1 6γ10 1 + 11γ10 1 6γ10 1 F 6γ10(δ1 γ1γ10) ∂t1 D − D D − D − − L
∂2F 2 2 2 = 2 2γ20 2 + βγ21γ32 1 ((2γ1 + γ10)γ21γ32β +2γ2,−1) 1 ∂t2 D − D D − D L 

2 2γ21 3 F + βγ21γ32( 1F ) − D D  2 +βγ γ (γ + γ γ δ )+2(γ γ + γ γ + γ γ − ) 21 32 1 10 1 − 1 21 3 20 2 1 2, 1

2 ∂ F β 3 2 3β = 1 3 γ32 1 + βγ32(γ1 +2γ10) 1 γ10γ32(2γ1 + γ10) 1 ∂t1∂t3 D D − 2 D D − 2 D L 

3β 2 2 3γ1,−1 2 3γ10 3 F + γ10γ32( 1F ) βγ32( 1F )( 1F ) − D − D 2 D − D D  3 2 + (2γ γ + βγ γ (γ + γ γ δ )+2γ − γ ). 2 10 3 32 10 1 10 1 − 1 1, 1 2 4.3.2 Jacobi ensemble Here, from the Virasoro constraints (2.1.35), one proceeds in the same way as before, by forming iF t=0 , i jF t=0, etc..., in terms of ti partials. For example, from the B | B 2B | expressions 1F t=0 , 1F , 0F t=0 , one extracts B− | B− t=0 B | ∂F ∂2F ∂F , , . ∂t ∂t2 ∂t 1 t=0 1 t=0 2 t=0

72 4, p.73 §

3 From the expressions 1F , 0 1F , 1F , and using the previous informa- t=0 − t=0 t=0 tion, one extracts B− B B | B | ∂F ∂2F ∂2F , , . ∂t ∂t3 ∂t ∂t 3 t=0 1 t=0 1 2 t=0 2 2 4 Finally, from the expressions 2F , 1 1F , 0F , 0 1F , 1F , t=0 − t=0 t=0 t=0 t=0 one deduces B | B B | B | B B− B−

∂4F ∂F ∂3F ∂2F ∂2F , , , , . (4.3.4) ∂t4 ∂t ∂t2∂t ∂t ∂t ∂t2 1 t=0 4 t=0 1 2 t=0 1 3 t=0 2 t=0

This provides all the partials, appearing in the KP equation (3.1.6).

4.3.3 Inserting partials into the integrable equation

From Theorem 3.3, the integrals In(t, c) , depending on β = 2, 1, 4, on t = (t1, t2, ...) and on the boundary points c =(c1, ..., c2r) of E, relate to τ-functions, as follows:

n ∞ i β tiz In(t, c) = ∆n(z) e 1 k ρ(zk)dzk En | | P Z kY=1   n!τn(t, c), n arbitrary, β =2 = n!τ (t, c), n even, β =1 , (4.3.5)  n  n!τ2n(t/2,c), n arbitrary, β =4

where τn(t, c) satisfies the KP-like equation

τn 2(t, c)τn+2(t, c) β n arbitrary for β =2 12 − δ =(KP ) log τ (t, c), (4.3.6) τ (t, c)2 1,4 t n n even for β =1, 4 n  with ∂ 4 ∂ 2 ∂2 ∂2 2 (KP ) F := +3 4 F +6 F . t ∂t ∂t − ∂t ∂t ∂t2  1   2  1 3 !  1 

Evaluating the left hand side of (4.3.6): Here In(t) will refer to the integral (4.3.5) over the full range. For β =2, the left hand side is zero. For β =1, the left hand side can be evaluated in terms of the probability Pn(E), as follows: taking into account Pn := Pn(E)= In(0,c)/In(0),

2 τn 2(t, c)τn+2(t, c) (n!) In 2(t, c)In+2(t, c) 12 − = 12 − τ (t, c)2 (n 2)!(n + 2)! I (t, c)2 n t=0 − n t=0 n(n 1) In 2(0)In+2(0) Pn 2P n+2 − − = 12 − 2 2 (n + 1)(n + 2) In(0) Pn

(1) Pn 2(E)Pn+2(E) − = 12bn 2 , Pn (E)

73 4, p.74 §

(1) 31 with bn given by

n(n 1) − 16b2 (Gaussian)  (1) n(n 1) In 2(0)In+2(0) n(n 1)(n+2a)(n+2a+1) b − − 4 n = − 2 =  16b (n + 2)(n + 1) In(0)   (Laguerre) Q (Jacobi) Q±  6  (4.3.7)  For β =4, we have:

2 τ2n 2(t/2,c)τ2n+2(t/2,c) (n!) In 1(t, c)In+1(t, c) 12 − = 12 − τ (t/2,c)2 (n 1)!(n + 1)! I (t, c)2 2n t=0 − n t=0 n In 1(0)In+1(0) Pn 1Pn+1 − − = 12 2 2 (n + 1) In(0) Pn

(4) Pn 1(E)Pn+1(E) − = 12bn 2 , Pn (E) with 2n(2n+1) 4b2 (Gauss) 2  (4) (n!) In 1(0)In+1(0) 2n(2n+1)(2n+a)(2n+a 1) b := − = 4 − n 2  b (n 1)!(n + 1)! In(0)  −  (Laguerre) Q (Jacobi) Q±  6  (4.3.8)  where Q is precisely the expression appearing on the left hand side of (4.1.10), and where Q6± is given by + for β =1 Q± = 3q (q +1)(q 3) q +4 4 q +1 (4.3.9) 6 − ± for β =4  −  p  (4) (1) The exact formulae bn and bn show they satisfy the duality property (4.1.2): a b b(4)(a, b, n)= b(1)( , , 2n). n n −2 −2 − 31this calculation is based on Selberg’s integrals (see Mehta [49], p 340). For instance, in the Jacobi case, one uses

n (β) β a b In = ∆n(x) (1 xj ) (1 + xj ) dxj n − [−1,1] j=1 Z Y − n 1 Γ(a + jβ/2 + 1)Γ(b + jβ/2 + 1)Γ((j + 1)β/2+1) = 2n(2a+2b+β(n−1)+2)/2 . Γ(β/2 + 1)Γ(a + b + (n + j 1)β/2+2) j=0 Y −

74 5, p.75 §

Evaluating the right hand side of (4.3.6): From section 2.4, it also follows that Fn(t; c)= log In(t; c) satisfies Virasoro constraints, corresponding precisely to the situation (4.3.1), with Gaussian ensemble32:

1 n γ1, 1 = 2 ,γ1,0 = γ1 =0, δ1 = 2 − − − n γ2, 1 =0,γ2,0 = 1/2,γ2,1 =0,γ2 = 4 σ1, δ2 =0  − 1 − 1 − n γ3, 1 = σ1,γ3,0 =0,γ3,1 = ,γ3,2 = γ3 =0, δ3 = σ1.  − − 4 − 2 − 4 33 Laguerre ensemble : δ1 = δ2 = δ3 = 0, and

n γ1, 1 =0,γ1,0 = 1,γ1 = 2 (σ1 + a), − − − n γ2, 1 =0,γ2,0 = σ1,γ2,1 = 1,γ2 = 2 σ1(σ1 + a),  − − − − n γ3, 1 =0,γ3,0 = σ1σ2,γ3,1 = σ2,γ3,2 = 1,γ3 = σ1σ2(σ1 + a).  − − − − − 2 Jacobi ensemble : see (4.3.4). They lead to expressions for

∂4F ∂2F ∂2F ∂2F , , , , ∂t4 ∂t2 ∂t ∂t ∂t2 1 t=0 2 t=0 1 3 t=0 1 t=0

in terms of k and k, which substituted in the right hand side of (4.3.6) - i.e. in the KP-expressionsD - leadsB to the right hand side of (4.1.4),(4.1.6), (4.1.9) and (4.1.10). In the Jacobi case, the right hand side of (4.3.6) contains the same coefficient 1/Q6± as in (4.3.9), which therefore cancels with the one appearing on the left hand side; see the 1,4 expression bn in (4.3.7) and (4.3.8).

5 Ensembles of infinite random matrices: Fredholm determinants, as τ-functions of the KdV equation

Infinite Hermitian matrix ensembles typically relate to the Korteweg-de Vries hierarchy, itself a reduction of the KP hierarchy; a brief sketch will be necessary. The KP-hierarchy 34 is given by tn-deformations of a pseudo-differential operator L: (commuting vector fields)

∂L n 1 ∂ = [(L )+, L], L = D + a 1D− + ..., with D = . (5.0.1) ∂tn − ∂x

32 Remember from section 2.1, σ1 = β(n 1) + 2. 33 − 3 Remember from section 2.1, σ1 = β(n 1)+ a + 2 and σ2 = β(n 2 )+ a + 3. 34 − − In this section, given P a pseudo-differential operator, P+ and P− denote the differential and the (strictly) smoothing part of P respectively.

75 5, p.76 §

+ Wave and adjoint wave functions are eigenfunctions Ψ (x, t; z) and Ψ−(x, t; z), depend- ing on x R, t C∞, z C, behaving asymptotically like (5.0.3) below and satisfying: ∈ ∈ ∈ + + + ∂Ψ n + zΨ = LΨ , =(L )+Ψ , ∂tn (5.0.2) ∂Ψ− n zΨ− = L⊤Ψ−, = (L⊤ )+Ψ−. ∂tn − + According to Sato’s theory, Ψ and Ψ− have the following representation in terms of a τ-function (see [25]): 1 (xz+ ∞ t zi) τ(t [z− ]) Ψ±(x, t; z) = e± 1 i ∓ P τ(t) (xz+ ∞ t zi) 1 = e± 1 i (1 + O(z− )), for z , (5.0.3) P ր∞ where τ satisfies and is characterized by the following bilinear relation

∞ ′ i (ti t )z 1 1 e 1 − i τ(t [z− ])τ(t′ + [z− ])dz =0, for all t, t′ C∞; (5.0.4) − ∈ I P the integral is taken over a small circle around z = . From the bilinear relation, one ∞ derives the KP-hierarchy, already mentioned in Theorem 3.1, of which the first equation reads as in (3.1.6). We consider the p-reduced KP hierarchy, i.e., the reduction to pseudo- differential L’s such that Lp = Dp + ... is a differential operator for some fixed p 2. Then (Lkp) = Lkp for all k 1 and thus ∂L/∂t = 0, in view of the deformation≥ equations + ≥ kp (5.0.1) on L. Therefore the variables tp, t2p, t3p, ... are not active and can thus be set = 0. The case p = 2 is particularly interesting and leads to the KdV equation, upon setting all even ti = 0. For the time being, take the integer p 2 arbitrary. The arbitrary linear combina- tions35 ≥

Φ±(x, t; z) := aω±Ψ±(x, t; ωz) (5.0.5) ω ζp X∈ p + p + p p are the most general solution of the spectral problems L Φ = z Φ and L⊤ Φ− = z Φ− respectively, leading to the definition of the kernels: x + E kx,t(y, z) := dx Φ−(x, t; y)Φ (x, t; z), and kx,t(y, z) := kx,t(y, z)IE(z), (5.0.6) Z where the integral is taken from a fixed, but arbitrary origin in R. In the same way E that Ψ±(x, t, z) has a τ-function representation, so also does kx,t(y, z) have a similar representation, involving the vertex operator:

+ Y (x, t; y, z) := aω−aω′ X(x, t; ωy,ω′z), (5.0.7) ′ ω,ω ζp X∈ 35ζ := ω such that ωp =1 p { } 76 5, p.77 §

where (see [25, 11, 12])

∞ 1 ∂ 1 (z y)x+ ∞(zi yi)t (y−i z−i) X(x, t; y, z) := e − 1 − i e 1 − i ∂ti . (5.0.8) z y P − P + − A condition aω aω = 0 is needed to guarantee that the right hand side of (5.0.7) is ω ζp ω free of singularities∈ in the positive quadrant y 0 and z 0 with i, j =1, ..., n and P { i ≥ j ≥ } limy z Y (x, t; y, z) exists. Indeed, using Fay identities and higher degree Fay identities, → one shows stepwise the following three statements, the last one being a statement about a Fredholm determinant36: 1 k (y, z) = Y (x, t; y, z)τ(t) x,t τ(t) 1 k det kx,t(yi, zj) = Y (x, t; yi, zi)τ 1 i,j n τ ≤ ≤ i=1   Y 1 τ(t, E) E λ E dz Y (x,t;z,z) det(I λkx,t) = e− τ =: . (5.0.9) − τ R τ(t) The kernel (5.0.12) at t = 0 will define the ststistics of a random Hermitian ensem- ble, when the size n . The next theorem is precisely a statement about Fredholm determinants of kernelsր∞ of the form (5.0.12); it will be identified at t = 0 with the prob- ability that no eigenvalue belongs to a subset E; see section 1.2. The initial condition that Virasoro annihilates τ0, as in sections 2.1.2 (Proof of Theorem 2.1), is now replaced by the initial condition (5.0.11) below.

(2) Theorem 5.1 (Adler-Shiota-van Moerbeke [11, 12]) Consider Virasoro generators Jℓ satisfying ∂ 1 zℓ+1Y (x, t; z, z)= J (2)(t),Y (x, t; z, z) , (5.0.10) ∂z 2 ℓ   where Y (x, t; z, z) is defined in (5.0.7), and a τ-function satisfying the Virasoro con- straint, with an arbitrary constant ckp:

J (2) c τ =0 for a fixed k 1. (5.0.11) kp − kp ≥ −   Then, given the disjoint union E R+, the Fredholm determinant of ⊂ E 1 kx,t(z, z′) p p Kx,t(λ,λ′) := p−1 p−1 IE(λ′), λ = z ,λ′ = z′ , (5.0.12) p z 2 z′ 2 36The Fredholm determinant of a kernel A(y,z) is defined by

∞ det(I λA)=1+ ( λ)m ... det (A(z ,z )) dz ...dz . − − i j 1≤i,j≤m 1 m m=1 X zZ1≤···≤Zzm

77 5, p.78 §

satisfies the following constraint for that same k 1: ≥ − 2r ∂ 1 ck+1 + (J (2) c ) τ det(I µKE )=0. (5.0.13) − i ∂c 2p kp − kp − x,t i=1 i ! X (2) The generators Jn take on the following precise form:

(1) ∂ Jn := + ( n)t−n ∂tn − J (2) := :J (1)J (1): (n + 1)J (1) n i j − n i+j=n X ∂2 ∂ = +2 it + it jt (n + 1)J (1). (5.0.14) ∂t ∂t i ∂t i j − n i+j=n i j −i+j=n j −i−j=n X X X

2 2 2 Remark: For KdV (i.e., p = 2), (L )⊤ = L = D q(x) holds, and thus the adjoint − + wave function has the simple expression: Ψ−(x, t; z)=Ψ (x, t; z). In the next two − examples, which deal with KdV, set

Ψ(x, t; z) := Ψ+(x, t; z).

Example 1: Eigenvalues of large random Hermitian matrices near the “soft edge” and the Airy kernel Remember from section 1.2, the spectrum of the Gaussian Hermitian matrix ensemble has, for large size n, its edge at √2n, near which the scaling is given by √2n1/6. ± Therefore, the eigenvalues in Theorem 5.2 must be expressed in that new scaling. Define r the disjoint union E = 1[c2i 1,c2i], with c2r possibly . ∪ − ∞ Theorem 5.2 Given the spectrum z z ... of the large random Hermitian matrix 1 ≥ 2 ≥ M, define the “eigenvalues” in the new scale:

2 zi ui =2n 3 1 for n . (5.0.15) √ n − ր∞  2  The probability of the “eigenvalues”

P (Ec) := P ( all eigenvalues u Ec) (5.0.16) i ∈ 2r k+1 ∂ 37 satisfies the partial differential equation (setting k := c ) B i=1 i ∂ci P 3 1 c 2 c 2 1 4( 0 ) 1 log P (E )+6( 1 log P (E )) =0. (5.0.17) B− − B − 2 B− B−   37When c = , that term in is absent. 2r ∞ Bk 78 5, p.79 §

In particular, the statistics of the largest “eigenvalue” u1 (in the new scale) is given by

∞ P (u x) = exp (α x)g2(α)dα , (5.0.18) 1 ≤ − −  Zx  with

3 g′′ = xg +2g (Painlev´eII) 3 2 2 . (5.0.19) e− 3 x ( g(x) = 1/4 for x . ∼ − 2√πx ր∞ The partial differential equation (5.0.17) is due to Adler-Shiota-van Moerbeke [11, 12]. The equation (5.0.19) for the largest eigenvalue is a special case of (5.0.17), but was first derived by Tracy-Widom [64], by methods of functional analysis. Proof: Remember from section 1.2, the statistics of the eigenvalues is governed by the Fredholm determinant of the kernel (1.2.4), for the Hermite polynomials. In the limit,

1 u v lim Kn √2n + , √2n + = K(u, v), n √2n1/6 √2n1/6 √2n1/6 ր∞   where

∞ ∞ iux x3/3 K(u, v)= A(x + u)A(x + v)dx, A(u)= e − dx. (5.0.20) 0 Z Z−∞ Then

P (Ec) := P ( all eigenvalue u Ec) = det(I K(u, v)I (v)). (5.0.21) i ∈ − E

In order to compute the PDE’s of this expression, with regard to the endpoints ci of the disjoint union E, one proceeds as follows: Consider the KdV wave function Ψ(x, t; z), as in (5.0.2), with initial condition:

1 2 xz+ 2 z3 1 2 Ψ(x, t ; z)= z 2 A(x + z )= e 3 (1 + O(z− )) , z , t = (0, 0, , 0, ...), 0 →∞ 0 3 (5.0.22) in terms of the Airy function38, which, by stationary phase, has the asymptotics:

3 3 1 ∞ y +yu 1 2 u 2 3 A(u) := e− 3 dy = u− 4 e 3 (1 + O(u− 2 )). √π Z−∞ The definition of A(u) is slightly changed, compared to (5.0.20). A(u) satisfies the differential equation A(y)′′ = yA(y), and thus the wave function Ψ(x, t0; z) satisfies 2 2 2 2 1 2 (D x)Ψ(x, t ; z) = z Ψ(x, t ; z). Therefore L = SD S− = D x, so − 0 0 |t=t0 |t=t0 − 38The i in the definition of the Airy function is omitted here.

79 5, p.80 §

that L2 is a differential operator, and Ψ is a KdV wave function, with τ(t) satisfying39 the Virasoro constraints (5.0.11) with c = 1 δ . The argument to prove these 2k − 4 k0 constraints is based on the fact that the linear span (a point in an infinite-dimensional Grassmannian)

n 2 z3 ∂ C − 3 √ = span ψn(z) := e z n A(u) , n =0, 1, 2, ... W ∂u 2  u=z 

2 1 ∂ 2 1 2 is invariant under multiplication by z and the operator 2z ( ∂z +2z ) 4 z− . 2 2 − Define for λ = z ,λ′ = z′ , the kernel Kt(λ,λ′),

1 ∞ Kt(λ,λ′) := 1 1 Ψ(x, t; z)Ψ(x, t; z′)dx, (5.0.23) 2z 2 z′ 2 Z0 which flows off the Airy kernel, by (5.0.22),

1 ∞ K (λ,λ′)= A(x + λ)A(x + λ′)dx. t0 2 Z0 Thus τ det(I KE ) satisfies (5.0.13), with that same constant c , for k = 1, 0, 1,... : − x,t kp − 2r ∂ 1 1 ck+1 + J (2) + δ τ det(I KE)=0. (5.0.24) − i ∂c 4 pk 16 k,0 − t i=1 i ! X

Upon shifting t3 t3 +2/3, in view of (5.0.22), the two first Virasoro constraints 7→ 2r k+1 ∂ for k = 1 and k =0 read: ( k := c ) − B i=1 i ∂ci

P 2 ∂ 1 ∂ t1 −1 log τ(t, E) = + iti log τ(t, E)+ B ∂t 2 ∂t −  4 1 ≥ i 2 Xi 3   ∂ 1 ∂ 1 log τ(t, E) = + it log τ(t, E)+ . (5.0.25) B0 ∂t 2 i ∂t  16 3 ≥ i Xi 1   The same method as in section 4 enables one to express all the t-partials, appearing in the KdV equation,

∂4 ∂2 ∂2 2 4 log τ(t, E)+6 log τ(t, E) =0, ∂t4 − ∂t ∂t ∂t2  1 1 3   1  39Although not used here, the τ-function is Kontsevich’s integral:((see [44, 10])

1 3 2 −Tr( 3 Y +Y Z) H dY e 1 −n 2 τ(t)= with tn = Tr(Z )+ δn,3,Z = diagonal matrix. −TrY 2Z −n 3 R H dY e R

80 5, p.81 §

in terms of c-partials, which upon substitution leads to the partial differential equation 3 1 2 1 4( 0 2 ) f + 6( 1f) = 0 (announced in (5.0.17)) for B− − B − B−
2r c ∂ c c E τ(t, E) f := 1 log P (E )= log P (E ), where P (E ) = det(I K )= . B− ∂c − τ(t) 1 i X When E =( , x), this PDE reduces to an ODE: −∞

2 d f ′′′ 4xf ′ +2f +6f ′ =0, with f = log P (max λi x). (5.0.26) − dx i ≤ According to Appendix on Chazy classes (section 9) , this equation can be reduced to

2 2 f ′′ +4f ′(f ′ xf ′ + f)=0, (Painlev´eII) (5.0.27) − which can be solved by setting

2 2 2 4 f ′ = g and f = g′ xg g . − − − 3 An easy computation shows g satisfies the equation g′′ = 2g + xg (Painlev´eII), thus leading to (5.0.19).

Example 2: Eigenvalues of large random Laguerre Hermitian matrices near the “hard edge” and the Bessel kernel Consider the ensemble of n n random matrices for the Laguerre probability distribu- × ν/2 z/2 tion, thus corresponding to (1.1.9) with ρ(dz) = z e− dz. Remember from section 1.2, the density of eigenvalues near the “hard edge” z = 0 is given by 4n for very large n. At this edge, the kernel (1.2.4) with Laguerre polynomials pn tends to the Bessel kernel [52, 30]:

1 1 (ν) u v (ν) 1 lim Kn , = K (u, v) := xJν (xu)Jν(xv)dx. (5.0.28) n 4n 4n 4n 2 0 ր∞   Z Therefore, the eigenvalues in the theorem below will be expressed in that new scaling. r Define, as before, the disjoint union E = 1[c2i 1,c2i]. ∪ −

Theorem 5.3 Given the spectrum 0 z1 z2 ... of the large random Laguerre- distributed Hermitian matrix M, define≤ the “eigenvalues”≤ ≤ in the new scale:

u =4nz for n . (5.0.29) i i ր∞ The statistics of the “eigenvalues”

P (Ec) := P ( all “eigenvalues” u Ec) (5.0.30) i ∈

81 5, p.82 §

c 2r k+1 ∂ leads to the following PDE for F = log P (E ): (setting k := c ) B i=1 i ∂ci 1 P 4 2 3 + (1 ν2) 2 + F 4( F )( 2F )+6( 2F )2 =0. (5.0.31) B0 − B0 − B0 B1 B0 − 2 − B0 B0 B0    In particular, for very large n, the statistics of the smallest eigenvalue is governed by x f(u) P (u x) = exp du , u 4nz , 1 ≥ − u 1 ∼ 1  Z0  with f satisfying

2 2 2 (xf ′′) 4 xf ′ f f ′ + (x ν )f ′ f f ′ =0. (Painlev´eV) (5.0.32) − − − − The equation (5.0.32) for
the smallest eigenvalue,
first derived by Tracy-Widom [64], by methods of functional analysis, is a special case of the partial differential equation (5.0.31), due to [11, 12]. Remark: This same theorem would hold for the Jacobi ensemble, near the “hard edges” z = 1. ± Proof: Define a wave function Ψ(x, t; z), flowing off

xz xz 1 Ψ(x, 0; z)= e B( xz)= e (1 + O(z− ), − where B(z) is the Bessel function40

z ν+1/2 ν+1/2 uz z e 2 ∞ z− e− 1 B(z)= ε√ze H (iz)= du =1+ O(z− ). ν Γ( ν +1/2) (u2 1)ν+1/2 − Z1 − As the operator ν2 1/4 L2 = D2 − |t=0 − x2 is a differential operator, we are in the KdV situation; again one may assume t2 = t4 = = 0 and we have ··· 1 + xz+ t zi τ(t [z− ]) Ψ−(x, t; z)=Ψ (x, t; z)= e i − , − P τ(t) in terms of a τ-function41 satisfying the following Virasoro constraints J (2)τ = ((2ν)2 1)δ τ. (5.0.33) 2k − k0 40 ε = i π/2 eiπν/2, 1/2 <ν< 1/2. 41 τ(t) is given by the− Adler-Morozov-Shiota-van Moerbeke double Laplace matrix transform, with p tn given in a similar way as in footnote 32 (see [10]):

ν−1/2 − Tr(Z2X) − Tr(XY 2) τ(t)= c(t) dX det X e dYS0(Y )e . + + ZHN ZHN

82 5, p.83 §

+ 1 iπν/2 + 1 iπν/2 Set p = 2, a1− = a 1 = 4π ie and a−1 = a1 = 4π e− in (5.0.5); this defines the − − 2 2 kernel (5.0.6) and so (5.0.12), which in terms of λ = z and λ′ = z′ , takes on the form:

x (ν) 1 iπν iπν K (λ,λ′) = ie 2 Ψ∗(x, t, z)+ e− 2 Ψ∗(x, t, z) x,t 4π√zz − · ′ Z  iπν iπν  e− 2 Ψ(x, t, z′)+ ie 2 Ψ(x, t, z′) dx, · −   which flows off the Bessel kernel,

x (ν) 1 K (λ,λ′) = xJ (x√λ)J (x√λ )dx. x,0 2 ν ν ′ Z0 J (√λ)√λ J ′ (√λ ) J (√λ )√λJ ′ (√λ) = ν ′ ν ′ − ν ′ ν for x =1. 2(λ λ ) − ′ The Fredholm determinant satisfies for E R and for k =0, 1,... : ⊂ + 2r ∂ 1 1 ck+1 + J (2) + ν2 δ τ det(I K(ν)E =0. (5.0.34) − i ∂c 4 2k 4 − k,0 − x,t 1 i ! X     Upon shifting t t + √ 1 and using the same as in (5.0.25), the equations 1 7→ 1 − Bi for k = 0 and k = 1 read

1 ∂ ∂ 1 1 log τ(t, E) = it + √ 1 log τ(t, E)+ ( ν2) B0 2 i ∂t − ∂t 4 4 − i 1 i 1 ! X≥ 1 ∂ 1 ∂2 ∂ 1 ∂ log τ(t, E) = it + + √ 1 + log τ(t, E). B1 2 i ∂t 2 ∂t2 − ∂t 2 ∂t i 1 i+2 1 3 1 ! X≥ (5.0.35)

Expressing the t-partials (5.0.20), appearing in the KdV-equation at t = 0 (see formula below (5.0.25)) in terms of the c-partials applied to log τ(0, E), leads to the following PDE for F = log P (Ec):

1 4 2 3 + (1 ν2) 2 + F 4( F )( 2F )+6( 2F )2 =0. (5.0.36) B0 − B0 − B0 B1 B0 − 2 − B0 B0 B0    Specializing this equation to the interval E = (0, x)leads to an ODEfor f := x ∂F/∂x, − namely

2 1 6 2 4 (x ν ) 1 f ′′′ + f ′′ f ′ + ff ′ + − f ′ f =0, (5.0.37) x − x x2 x2 − 2x2 which is an equation of the type (9.0.1); changing x y x and f y f leads again to an equation of type (9.0.1), with P (x)= x, 4Q(x)= x− ν2 and R =− 0. According to − − 83 6, p.84 §

Cosgrove-Scoufis [24] (see the Appendix on Chazy classes), this equation can be reduced to the equation (9.0.2), with the same P, Q, R and with c = 0. Since P (x) = x, this equation is already in one of the canonical forms (9.0.3), which upon changing back x and f, leads to

2 2 2 (xf ′′) +4 xf ′ + f f ′ + (x ν )f ′ f f ′ =0. (Painlev´eV) − − −

Example 3: Eigenvalues of large random Gaussian Hermitian matrices in the bulk and the sine kernel Setting ν = 1/2 yields kernels related to the sine kernel: ± 1 x sin xy sin xz K(+1/2)(y2, z2) = dx x,0 π y1/2z1/2 Z0 1 sin x(y z) sin x(y + z) = − 2π y z − y + z  −  x ( 1/2) 2 2 1 cos xy cos xz K − (y , z ) = dx x,0 π y1/2z1/2 Z0 1 sin x(y z) sin x(y + z) = − + . 2π y z y + z  −  Therefore the sine-kernel obtained in the context of the bulk-scaling limit (see 1.2.7) (+1/2) ( 1/2) is the sum Kx,0 +Kx,−0 . Expressing the Fredholm determinant of this sum in terms of the Fredholm determinants of each of the parts, leads to the Painlev´eV equation (1.2.8).

6 Coupled random Hermitian ensembles

Consider a product ensemble (M , M ) 2 := of n n Hermitean matrices, 1 2 ∈ Hn Hn × Hn × equipped with a Gaussian probability measure,

1 2 2 Tr(M +M 2cM1M2) cndM1dM2 e− 2 1 2 − , (6.0.1) where dM dM is Haar measure on the product 2 , with each dM , 1 2 Hn i n n 2 2 dM1 = ∆n(x) dxidU and dM2 = ∆n(y) dyidU (6.0.2) 1 1 Y Y decomposed into radial and angular parts. In terms of the coupling constant c, appear- ing in (6.0.1), and the boundary of the set

r s R2 E = E1 E2 := i=1[a2i 1, a2i] i=1[b2i 1, b2i] , (6.0.3) × ∪ − × ∪ − ⊂

84 7, p.85 §

define differential operators , of “weight” k, Ak Bk 1 r ∂ s ∂ 1 r ∂ s ∂ = + c = c + A1 c2 1 ∂a ∂b B1 1 c2 ∂a ∂b − 1 j 1 j ! − 1 j 1 j ! r ∂ X ∂ X s ∂ X ∂ X = a c = b c , A2 j ∂a − ∂c B2 j ∂b − ∂c j=1 j j=1 j X X forming a closed Lie algebra42. The following theorem follows, via similar methods, from the Virasoro constraints (3.3.5) and the 2-Toda equation (3.3.6):

Theorem 6.1 (Gaussian probability) (Adler-van Moerbeke [3]) The joint statistics

1 2 2 Tr(M +M 2cM1M2) dM1dM2 e− 2 1 2 − 2 2 n(E1 E2) Pn(M n(E1 E2)) = Z ZH × 1 2 2 ∈ H × Tr(M +M 2cM1M2) dM1dM2 e− 2 1 2 − 2 Z ZHn n 1 2 2 (x +y 2cxkyk) ∆n(x)∆n(y) e− 2 k k− dxkdyk n Z ZE k=1 = Yn 1 2 2 (x +y 2cxkyk) ∆n(x)∆n(y) e− 2 k k− dxkdyk R2n Z Z Yk=1 satisfies the non-linear third-order partial differential equation43 (independent of n) 1 ( Fn := n log Pn(E) ): c c F , F + F , F + =0. (6.0.4) 2 1 n 1 1 n c2 1 2 1 n 1 1 n c2 1 B A B A 1 − A B A B 1  − A  − B 7 Random permutations

The purpose of this section is to show that the generating function of the probability 1 P (L(π ) ℓ)= # π S L(π ) ℓ n ≤ n! { n ∈ n n ≤ } 42

1+ c2 2c [ , ] = 0 [ , ]= [ , ]= A1 B1 A1 A2 1 c2 A1 A2 B1 1 c2 A1 − − 2c 1+ c2 [ , ] = 0 [ , ]= − [ , ]= . A2 B2 A1 B2 1 c2 B1 B1 B2 1 c2 B1 − −

43 in terms of the Wronskian f,g X = Xf.g f.Xg, with regard to a first order differential operator X. { } −

85 7, p.86 §

is closely related to a special solution of the Painlev´eV equation, with peculiar initial condition. Remember from section 1.4, L(πn) is the length of the longest increasing sequence in the permutation πn.

Theorem 7.1 For every ℓ 0, ≥ ∞ xn x x P (L(π ) ℓ)= e√x tr(M+M¯ )dM = exp log g (u)du, n! n ≤ u ℓ n=0 U(ℓ) 0 X Z Z   (7.0.1)

with gℓ satisfying the initial value problem for Painlev´eV:

2 2 g′ 1 1 g′ 2 ℓ g 1 g′′ + + + g(g 1) − =0 − 2 g 1 g u u − − 2u2 g    − (7.0.2)   uℓ  g (u)=1 + O(uℓ+1), near u =0. ℓ − (ℓ!)2   The Painlev´eV equation for this integral was first found by Tracy-Widom [68]. This systematic derivation and the initial condition are due to Adler-van Moerbeke [7].

Proof: Upon inserting (t1, t2, ...) and (s1,s2, ...) variables in the U(n)-integral (7.0.1), the integral

∞ i i Tr (tiM siM¯ ) In(t, s) = e 1 − dM (7.0.3) ZU(n) P ∞ i −i k ℓ (tiz siz ) dz = n! det z − e 1 − = n!τn(t, s) S1 P 2πiz 0 k,ℓ n 1 Z  ≤ ≤ −

puts us in the conditions of Theorem 3.5. It deals with semi-infinite matrices L1 and 1 hL2⊤h− of “rank 2”, having diagonal elements

∂ τn ∂ τn 1 bn := log =(L1)n 1,n 1 and bn∗ := log =(hL2⊤h− )n 1,n 1. ∂t1 τn 1 − − −∂s1 τn 1 − − − −

To summarize Theorem 3.5, In(t, s) satisfies the following three types of identities:

86 7, p.87 §

(i) Virasoro: (F := log τn) (see (3.4.7))

−1τn ∂ ∂ ∂ 0= V = (i + 1)t (i 1)s − + n F + nt τ  i+1 ∂t − − i 1 ∂s ∂s  1 n ≥ i ≥ i 1 Xi 1 Xi 2 τ  ∂ ∂  0= V0 n = it is F τ i ∂t − i ∂s n ≥ i i Xi 1  

1τn ∂ ∂ ∂ 0= V = (i + 1)s + (i 1)t − F + ns . τ − i+1 ∂s − i 1 ∂t ∂t  1 n ≥ i ≥ i 1 Xi 1 Xi 2   2 2 2 ∂ −1τn ∂ ∂ ∂ 0= V = (i + 1)t (i 1)s − + n F + n ∂t τ  i+1 ∂t ∂t − − i 1 ∂t ∂s ∂t ∂s  1 n ≥ 1 i ≥ 1 i 1 1 Xi 1 Xi 2   (7.0.4)

(ii) two-Toda: (see (3.4.8))

2 2 3 ∂ log τn ∂ τn ∂ ∂ = 2 log log τn 2 log τn ∂s2∂t1 − ∂s1 τn 1 ∂s1∂t1 − ∂s1∂t1 − ∂2 ∂3 = 2bn∗ log τn 2 log τn, (7.0.5) ∂s1∂t1 − ∂s1∂t1 (iii) Toeplitz: (see (3.4.11))

∂ τn ∂ τn (τ)n = log log T ∂t1 τn−1 ∂s1 τn−1 ∂2 ∂2 ∂ ∂ τ + 1+ log τ 1+ log τ log n ∂s ∂t n ∂s ∂t n − ∂s ∂t τ  1 1   1 1 1  1 n−1  ∂2 ∂2 ∂ = b b∗ + 1+ log τ 1+ log τ b =0. − n n ∂s ∂t n ∂s ∂t n − ∂s n  1 1   1 1 1  (7.0.6)

Defining the locus = all ti = si = 0, except t1,s1 = 0 , and using the second relation (7.0.4), we haveL on{ , 6 } L

0τn ∂ ∂ V = t1 s1 log τn =0, τn ∂t1 − ∂s1 L   L

implying τ (t, s) is a function of x := t s only. Therefore we may write τ = n − 1 1 n L ∂ ∂ ∂ ∂ ∂2 ∂ ∂ L τn(x), and so, along , we have = s1 , = t1 , = x . Setting ∂t1 ∂x ∂s1 ∂x ∂t1∂s1 ∂x ∂x L − − − ∂ ∂ ∂2 fn(x)= x log τn(x)= log τn(t, s) , (7.0.7) ∂x ∂x −∂t1∂s1 L

87 8, p.88 §

and using x = t s , the two-Toda relation (7.0.5) takes on the form − 1 1 2 2 2 ∂ log τn ∂ ∂ ∂ log τn s1 = s1 2bn∗ log τn ∂s2∂t1 ∂s1∂t1 − ∂s1 ∂s1∂t1 L    bn∗ = x(2 fn + fn′ ). t1 Setting relation (7.0.7) into the Virasoro relations (7.0.4) yields

0τn 0τn 1 ∂ ∂ τn 0= V V − = t1 s1 log = t1bn + s1bn∗ τn − τn 1 ∂t1 − ∂s1 τn 1 − L   − L 2 2 ∂ 1τn ∂ ∂ 0= V− = s1 + n log τn + n ∂t1 τn − ∂s2∂t1 ∂t1∂s1 L   L

bn∗ = x 2 f (x)+ f ′ (x) + n( f (x)+1). − t n n − n  1 

This is a system of two linear relations in bn and bn∗ , whose solution, together with its derivatives, are given by:

2 bn∗ bn n(fn 1) + xfn′ ∂bn ∂ bn x(fnfn′′ fn′ )+(fn + n)fn′ = = − , = x = − 2 . t1 −s1 − 2xfn ∂s1 ∂x s1 2fn Substituting the result into the Toeplitz relation (7.0.6), namely ∂ b b∗ = (1 f ) 1 f b , n n − n − n − ∂s n  1  leads to fn satisfying Painlev´eequation (7.0.2), with g = fn, as in (7.0.7) and u = x. Note, along the locus , we may set t = √x and s = √x, since it respects t, s = x. L 1 1 − 1 − Thus, In(t, s) equals (7.0.1). The initial|L condition (7.0.2) follows from the fact that as long as 0 n ℓ, the ≤ ≤ inequality L(π ) ℓ is always verified, and so n ≤ ℓ ∞ xn xn xℓ+1 # π S L(π ) ℓ = + ((ℓ + 1)! 1) + O(xℓ+2) (n!)2 { ∈ n n ≤ } n! (ℓ + 1)!2 − 0 0 X X xℓ+1 = exp x + O(xℓ+2) , − (ℓ + 1)!2   thus proving Proposition 7.1.

Remark: Setting g(x) f (x)= n g(x) 1 − leads to standard Painlev´eV, with α = δ =0, β = n2/2,γ = 2. − − 88 8, p.89 § 8 Random involutions

This section deals with a generating function for the distribution of the length of the 0 longest increasing sequence of a fixed-point free random involution π2k, with the uniform distribution: 2kk! P L(π0 ) ℓ +1, π0 S0 = # π0 S0 L(π0 ) ℓ +1 . 2k ≤ 2k ∈ 2k (2k)! { 2k ∈ 2k 2k ≤ }

Proposition 8.1 (Adler-van Moerbeke [7]) The generating function

∞ (x2/2)k 2 P (L(π0 ) ℓ + 1) k! 2k ≤ k=0 X xTrM xTrM = EO(ℓ+1)− e + EO(ℓ+1)+ e x x + f −(u) f (u) = exp ℓ du + exp ℓ du , (8.0.1) u u Z0  Z0  where f = fℓ±, satisfies the initial value problem:

2 2 2 1 6 2 4 16u + ℓ 16 2(ℓ 1) f ′′′ + f ′′ + f ′ ff ′ f ′ + f + − =0 u u − u2 − u2 u u   ℓ+1  2 u ℓ+2  f ±(u)= u + O(u ), near u =0. (Painlev´eV) ℓ ± ℓ!  (8.0.2)  Proof: The first equality in (8.0.1) follows immediately from proposition 1.1. The results of section 1.3 lead to

n xTrM x 2 2xzk a b e dM = e± ∆n(z) e (1 zk) (1 + zk) dzk, (8.0.3) n O(2n+1)± [ 1,1] − Z Z − Yk=1

with a = 1/2, b = 1/2, (with corresponding signs). Inserting ti’s in the integral, the ± ∓ x perturbed integral, with e± removed and with t1 =2x, reads

n ∞ i 2 a b tiz In(t)= ∆n(z) (1 zk) (1 + zk) e 1 k dzk = n!τn(t); (8.0.4) [ 1,1]n − P Z − kY=1 this is precisely integral (3.1.4) of section 3.1.1 and thus it satisfies the Virasoro con- straints (3.1.5), but without boundary contribution F . Explicit Virasoro expressions Bi appear in (2.1.35), upon setting β = 2. Also, τn(t), as in (3.1.4), (see Theorem 3.1)

89 8, p.90 § satisfies the KP equation (3.1.6). Differentiating the Virasoro constraints in t1 and t2, and restricting to the locus

:= t = x, all other t =0 , L { 1 i } lead to a linear system of five equations, with b = a b, b = a + b, 0 − 1 1 J(2) J(2) + b J(1) + b J(1) I =0, k = 1, 0 I k+2 − k 0 k+1 1 k+2 n − n L ∂ 1  J(2) J(2) + b J(1) + b J(1) I =0, k = 1, 0 ∂t I k+2 − k 0 k+1 1 k+2 n − 1 n L ∂ 1   J(2) J(2) J(1) J(1) k+2 k + b0 k+1 + b1 k+2 In =0, k = 1 ∂t2 In − −   L in five unknowns (Fn = log τn) ∂F ∂F ∂2F ∂2F ∂2F n , n , n , n , n . ∂t ∂t ∂t ∂t ∂t ∂t ∂t2 2 3 1 2 1 3 2 L L L L L

Setting t1 = x and F n′ = ∂Fn/∂x , the solution is given by the following expressions, ∂F 1 n = (2n + b )F ′ + n(b x) ∂t −x 1 n 0 − 2 L   ∂F 1 n = x F ′′ + F ′2 + (b x)F ′ + n(n + b ) (2n + b ) ((2n + b )F ′ + b n) ∂t −x2 n n 0 − n 1 − 1 1 n 0 3 L 
 ∂2F 1 n = (2n + b )(xF ′′ F ′ ) b n ∂t ∂t −x2 1 n − n − 0 1 2 L 2   ∂ Fn 1 2 ′′′ ′ ′′ 2 ′′ ′2 ′ 2 ′′ = x (F +2F F ) x (x b0x + 1)F + F + b0F + (2n + b1) F ∂t ∂t −x3 n n n − − n n n n 1 3 L  +n(n + b )) + 2(2n + b )2F ′ +2b n(2n + b ) 1 1 n 0 1  ∂2F 1 n = x 2F ′2 +2b F ′ + ((2n + b )2 + 2)F ′′ +2n(n + b ) ∂t2 x3 n 0 n 1 n 1 2 L 
3(2n + b )2F ′ 3b n(2n + b ) . − 1 n − 0 1  Putting these expressions into KP and setting t1 = x, one finds:

∂ 4 ∂ 2 ∂2 ∂2 2 0 = +3 4 F +6 F ∂t ∂t − ∂t ∂t n ∂t2 n  1   2  1 3 !  1  1 3 2 2 2 2 = x F ′′′′ +4x F ′′′ + x 4x +4b x +2 (2n + b ) F ′′ +8x F ′F ′′ x3 − 0 − 1 3 2 2 2 2 +6x F ′′ +2xF ′ + 2b x (2n + b ) F ′ + n(2x b )(
n + b ) b n . 0 − 1 − 0 1 − 0


90 9, p.91 §

d d Finally, the function H(x) := x dx F (x)= x dx log τn(x) satisfies

2 2 2 2 x H′′′ + xH′′ +6xH′ 4H +4x 4bx + (2n + a) H′ + (4x 2b)H − − − +2n(n + a)x bn(2n + a)=0. −
(8.0.5)

This 3rd order equation is Cosgrove’s [24, 23] equation, with P = x, 4Q = 4x2 + − 4bx (2n + a)2, 2R = 2n(n + a)x bn(2n + a). So, this 3rd order equation can be transformed− into the Painlev´eV equation− (9.0.3) in the appendix. The boundary condition f(0) = 0 follows from the definition of H above, whereas, after an elementary, but tedious computation, f ′(0) = f ′′(0) = 0 follows from the differential equation (8.0.5) and the Aomoto extension [15] (see Mehta [49], p. 340) of Selberg’s integral:44

1 1 β n γ δ m ... x1 ...xm ∆(x) xj (1 xj ) dx1...dxn γ +1+(n j)β/2 0 0 | | j=1 − = − . 1 1 β n γ δ γ + δ +2+(2n j 1)β/2 R R ... ∆(x) j=1Qxj (1 xj ) dx1...dxn j=1 0 0 | | − Y − − However,R theR initial conditionQ (8.0.2) is a much stronger statement, again stemming from the fact that as long as 0 n ℓ, the inequality L(πn) ℓ is trivially verified, thus leading to ≤ ≤ ≤

x2 xℓ+1 E exTrM = exp + O(xℓ+2) , O±(ℓ+1) 2 ± (ℓ + 1)!   ending the proof of Proposition 8.1.

9 Appendix: Chazy classes

Most of the differential equations encountered in this survey belong to the general Chazy class

f ′′′ = F (z,f,f ′, f ′′), where F is rational in f, f ′, f ′′ and locally analytic in z, subjected to the requirement that the general solution be free of movable branch points; the latter is a branch point whose location depends on the integration constants. In his classification Chazy found thirteen cases, the first of which is given by

P ′ 6 2 4P ′ P ′′ 2 4Q 2Q′ 2R f ′′′ + f ′′ + f ′ ff ′ + f + f ′ f + = 0 (9.0.1) P P − P 2 P 2 P 2 − P 2 P 2 1 Reγ +1 Reδ +1 44where Reγ, Reδ > 1, Reβ > 2 min , , − − n n 1 n 1  − − 

91 9, p.92 §

with arbitrary polynomials P (z), Q(z), R(z) of degree 3, 2, 1 respectively. Cosgrove and Scoufis [24, 23], (A.3), show that this third order equation has a first integral, which is second order in f and quadratic in f ′′,

2 4 2 2 f ′′ + (P f ′ + Qf ′ + R)f ′ (P ′f ′ + Q′f ′ + R′)f P 2 −  1 2 1 3 + (P ′′f ′ + Q′′)f P ′′′f + c = 0; (9.0.2) 2 − 6  c is the integration constant. Equations of the general form

2 f ′′ = G(x, f, f ′) are invariant under the map a z + a a f + a z + a x 1 2 and f 5 6 7 . 7→ a3z + a4 7→ a3z + a4 Using this map, the polynomial P (z) can be normalized to

P (z)= z(z 1), z, or 1. − In this way, Cosgrove shows (9.0.2) is a master Painlev´eequation, containing the 6 Painlev´eequations. In each of the cases, the canonical equations are respectively:

2 3 g′′ = 4g′ 2g′(zg′ g)+ A (Painlev´eII) • − − − 1 2 3 2 g′′ = 4g′ + 4(zg′ g) + A g′ + A (Painlev´eIV) • − − 1 2 2 2 (zg′′) =(zg′ g) 4g′ + A (zg′ g)+ A + A g′ + A • − − 1 − 2 3 4   (Painlev´eV)

2 2 2 (z(z 1)g′′) =(zg′ g) 4g′ 4g′(zg′ g)+ A + A g′ + A g′ + A • − − − − 2 1 3 4   (Painlev´eVI) (9.0.3)

Painlev´eII equation above can be solved by setting

1 2 1 2 z 2 ε1 g(z) = (u′) (u + ) (α + )u 2 − 2 2 − 2 ε1 1 2 z g′(z) = u′ (u + ) − 2 − 2 2 1 z A = (α +(u2 + )2ε )2, (ε = 1). 1 4 2 1 ±

92 9, p.93 §

Then u(z) satisfies yet another version of the Painlev´eII equation

3 u′′ =2u + zu + α. (Painlev´eII)

Now, each of these Painlev´eII,IV,V,VI equations can be transformed into the standard Painlev´eequations, which are all differential equations of the form

f ′′ = F (z,f,f ′), rational in f, f ′, analytic in z, whose general solution has no movable critical points. Painlev´eshowed that this re- quirement leads to 50 types of equations, six of which cannot be reduced to known equations.

References

[1] M. Adler and P. van Moerbeke: B¨acklund transformations, Birkhoff strata and Isospectral sets of differential operators, Advances in Mathematics, 108, 140-204, (1994).

[2] M. Adler and P. van Moerbeke: Matrix integrals, Toda symmetries, Virasoro con- straints and orthogonal polynomials, Duke Math. J. 80, 863-911 (1995).

[3] M. Adler and P. van Moerbeke: The spectrum of coupled random matrices, Annals of Mathematics, 149, 921–976 (1999).

[4] M. Adler and P. van Moerbeke: String orthogonal Polynomials, String Equations and two-Toda Symmetries, Comm. Pure and Appl. Math., L 241–290 (1997).

[5] M. Adler and P. van Moerbeke: Vertex operator solutions to the discrete KP- hierarchy, Comm. Math. Phys., 203 185–210 (1999).

[6] M. Adler and P. van Moerbeke: Hermitian, symmetric and symplectic random ensembles: PDE’s for the distribution of the spectrum, Annals of Mathematics, ( Sept 2000). (solv-int/9903009 and math-ph/0009001)

[7] M. Adler and P. van Moerbeke: Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math., (2000) (math.CO/9912143 )

[8] M. Adler and P. van Moerbeke: The Pfaff lattice, matrix integrals and a map from Toda to Pfaff, Duke Math J. (2000) (solv-int/9912008)

[9] M. Adler, E. Horozov and P. van Moerbeke: The Pfaff lattice and skew-orthogonal polynomials, International Mathematics Research notices, 11, 569-588 (1999).

93 9, p.94 §

[10] M. Adler, A. Morozov, T. Shiota and P. van Moerbeke: A matrix integral solution to [P, Q] = P and matrix Laplace transforms, Comm. Math. Phys., 180, 233-263 (1996).

[11] M. Adler, T. Shiota and P. van Moerbeke: Random matrices, vertex operators and the Virasoro algebra, Phys. Lett. A 208, 67-78, (1995).

[12] M. Adler, T. Shiota and P. van Moerbeke: Random matrices, Virasoro algebras and non-commutative KP, Duke Math. J. 94, 379-431 (1998).

[13] M. Adler, T, Shiota and P. van Moerbeke: Pfaff τ-functions , Math. Annalen (2000) (solv-int/9909010)

[14] D. Aldous and P. Diaconis: Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem, Bull. Am. Math. Soc. (new series) 36 (4), 413–432 (1999).

[15] K. Aomoto: Jacobi polynomials associated with Selberg integrals , SIAM J. Math. Anal. 18 , 545–549 (1987).

[16] H. Awata, Y. Matsuo, S. Odake and J. Shiraishi: Collective field theory, Calogero- Sutherland Model and generalized matrix models , RIMS-997 reprint , (1994). (hep- th/9411053)

[17] B. Baik, P. Deift and K. Johansson: On the distribution of the length of the longest increasing subsequence of random permutations, Math. Archive, Journal Amer. Math. Soc. 12 , 1119-1178 (1999) (MathCO/9810105).

[18] J. Baik and E. Rains: Algebraic aspects of increasing subsequences, math.CO/9905083 (1999).

[19] P. Bleher, A. Its: Semiclassical asymptotics of orthogonal polynomials, Riemann- Hilbert problem and universality in the matrix model, Ann. of Math. 150 1–81 (1999).

[20] N. Bourbaki:Alg`ebre de Lie, Hermann, Paris.

[21] Bowick and E. Br´ezin: Universal scaling of the tail of the density of eigenvalues in random matrix models , Phys. Letters B 268, 21–28 (1991).

[22] E. Br´ezin, H. Neuberger: Multicritical points of unoriented random surfaces, Nu- clear Physics B 350, 513–553 (1991).

[23] C. M. Cosgrove: Chazy classes IX-XII of third-order differential equations, Stud. Appl. Math. 104 ,3, 171–228 (2000).

94 9, p.95 §

[24] C. M. Cosgrove, G. Scoufis: Painlev´eclassification of a class of differential equa- tions of the second order and second degree , Studies. Appl. Math. 88 , 25–87 (1993).

[25] E. Date, M. Jimbo, M. Kashiwara, T. Miwa: Transformation groups for soliton equations, In: Proc. RIMS Symp. Nonlinear integrable systems — Classical and quantum theory (Kyoto 1981), pp. 39–119. Singapore : World Scientific 1983.

[26] P. Diaconis, M. Shashahani: On the eigenvalues of random matrices J. Appl. Prob., suppl. in honour of Tak`acs 31A, 49-61 (1994).

[27] F. Dyson: Statistical theory of energy levels of complex systems, I, II and III, J. Math Phys 3 140–156, 157–165, 166–175 (1962).

[28] P. Erd¨os, G. Szekeres: A combinatorial theorem in geometry, Compositio Math., 2, 463–470 (1935).

[29] A.S. Fokas, A.R. Its, A.V. Kitaev: The isomonodromy approach to matrix models in 2d quantum gravity, Comm. Math. Phys., 147, 395–430 (1992).

[30] P.J. Forrester: The spectrum edge of random matrix ensembles , Nucl. Phys. B, 402, 709-728 (1993).

[31] I. M. Gessel: Symmetric functions and P-recursiveness , J. of Comb. Theory, Ser A, 53, 257–285 (1990)

[32] L. Haine, J.P. Semengue: The Jacobi polynomial ensemble and the Painlev´eVI equation, J. of Math. Phys., 40, 2117–2134 (1999).

[33] J.M. Hammersley: A few seedlings of research, Proc. Sixth. Berkeley Symp. Math. Statist. and Probability, Vol. 1, 345–394, University of California Press (1972).

[34] Harish-Chandra: Differential operators on a semi-simple Lie algebra Amer. J. of Math., 79, 87–120 (1957).

[35] S. Helgason: Groups and geomatric analysis; integral geometry, invariant differen- tial operators, and spherical functions, Acad. Press 1984

[36] S. Helgason: Differential geometry and symmetric spaces, Acad. Press 1962

[37] M. Hisakado: Unitary matrix models and Painlev´eIII , Mod. Phys. Letters, A 11 3001–3010 (1996).

[38] Cl. Itzykson, J.-B. Zuber: The planar approximation , J. Math. Phys. 21, 411–421 (1980).

95 9, p.96 §

[39] M. Jimbo, T. Miwa, Y. Mori and M. Sato: Density matrix of an impenetrable Bose gas and the fifth Painlev´etranscendent, Physica 1D, 80–158 (1980).

[40] K. Johansson: On random matrices from the compact classical groups, Ann. of Math., 145, 519–545 (1997)

[41] K. Johansson: The Longest Increasing Subsequence in a Random Permutation and a Unitary Random Matrix Model, Math. Res. Lett., 5, no. 1-2, 63–82 (1998)

[42] V.G. Kac and J. van de Leur: The geometry of spinors and the multicomponent BKP and DKP hierarchies in ”The bispectral problem (Montreal PQ, 1997)”, CRM Proc. Lecture notes 14, AMS, Providence, 159-202 (1998).

[43] R.D. Kamien, H.D. Politzer, M.B. Wise: Universality of random-matrix predictions for the statistics of energy levels Phys. rev. letters 60, 1995–1998 (1988).

[44] M. Kontsevich: Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147, 1-23 (1992).

[45] D. Knuth: “The Art of Computer programming, Vol III: Searching and Sorting”, Addison-Wesley, Reading, MA, 1973.

[46] B.F. Logan and L.A. Shepp: A variational problem for random Young tableaux, Advances in Math., 26, 206–222 (1977).

[47] I.G. MacDonald: “Symmetric functions and Hall polynomials”, Clarendon Press, 1995.

[48] G. Mahoux, M.L. Mehta: A method of integration over matrix variables: IV J. Phys. I (France) 1, 1093–1108 (1991).

[49] M.L. Mehta: Random matrices, 2nd ed. Boston: Acad. Press, 1991.

[50] M.L. Mehta: Matrix Theory, special topics and useful results, Les ´editions de Physique, Les Ulis, France 1989.

[51] Moore, G.: Matrix models of 2D gravity and isomonodromic deformations, Progr. Theor. Phys., Suppl. 102, 255–285 (1990).

[52] T. Nagao, M. Wadati: Correlation functions of random matrix ensembles related to classical orthogonal polynomials, J. Phys. Soc of Japan, 60 3298–3322 (1991).

[53] . A.M. Odlyzko: On the distribution of spacings between zeros of the zeta function, Math. Comput. 48 273-308 (1987).

[54] A. Okounkov: Random matrices and random permutations, Math.CO/99-03176, (1999).

96 9, p.97 §

[55] L.A. Pastur: On the universality of the level spacing distribution for some ensem- bles of random matrices, Letters Math. Phys., 25 259–265 (1992). [56] C.E. Porter and N. Rosenzweig, Statistical properties of atomic and nuclear spectra, Ann. Acad. Sci. Fennicae, Serie A, VI Physica 44, 1-66 (1960) Repulsion of energy levels in complex atomic spectra, Phys. Rev 120, 1698–1714 (1960). [57] E. M. Rains: Topics in Probability on compact Lie groups, Harvard University doctoral dissertation, (1995). [58] E. M. Rains: Increasing subsequences and the classical groups , Elect. J. of Com- binatorics, 5, R12, (1998). [59] P. Sarnak: MSRI-proceedings, (2000). [60] P. Sarnak: Arithmetic quantum chaos, Israel Math. Conf. Proceedings, 8, 183–236 (1995). [61] A. Terras: “Harmonic analysis on Symmetric Spaces and Applications II”, Springer-verlag, 1988. [62] C.L. Terng: Isoparametric submanifolds and their Coxeter groups, J. Differential Geometry. 21 79–107 (1985), . [63] C.L. Terng, W. Y. Hsiang and R. S. Palais: The topology of isoparametric sub- manifolds in Euclidean spaces, J. of Diff. Geometry, 27, 423–460 (1988). [64] C.A. Tracy and H. Widom: Level-Spacings distribution and the Airy kernel, Com- mun. Math. Phys., 159, 151–174 (1994). [65] C.A. Tracy and H. Widom: Level spacing distributions and the Bessel kernel, Commun. Math. Phys., 161, 289–309 (1994). [66] C.A. Tracy and H. Widom: Fredholm Determinants, Differential Equations and Matrix Models, Comm. Math. Phys., 163, 33-72 (1994). [67] C.A. Tracy and H. Widom: On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177, 103–130 (1996). [68] C.A. Tracy and H. Widom: Random unitary matrices, permutations and Painlev´e, math.CO /9811154 , (1999). [69] K. Ueno and K. Takasaki: Toda Lattice Hierarchy, Adv. Studies in Pure Math. 4, 1–95 (1984). [70] S.M. Ulam: Monte Carlo calculations in problems of mathematical physics, in Modern Mathematics for the Engineers, E.F. Beckenbach ed., McGraw-Hill, 261– 281 (1961).

97 9, p.98 §

[71] J. van de Leur: Matrix Integrals and Geometry of Spinors (solv-int/9909028)

[72] P. van Moerbeke: The spectrum of random matrices and integrable systems, Group 21, Physical applications and Mathematical aspects of Geometry, Groups and Al- gebras, Vol.II, 835–852, Eds.: H.-D. Doebner, W. Scherer, C. Schulte, World sci- entific, Singapore, 1997.

[73] P. van Moerbeke: Integrable foundations of string theory, in Lectures on Integrable systems, Proceedings of the CIMPA-school, 1991, Ed.: O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, World scientific, 163–267 (1994).

[74] A.M. Vershik and S.V. Kerov: Asymptotics of the Plancherel measure of the sym- metric group and the limiting form of Young tableaux, Soviet Math. Dokl., 18, 527–531 (1977).

[75] H. Weyl: The classical groups, Princeton University press, 1946.

[76] E.P. Wigner:On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambr. Phil. Soc. 47 790–798 (1951).

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