"Integrable Systems, Random Matrices and Random Processes"

Contents

Integrable Systems, Random Matrices and Random Processes Mark Adler ...... 1 1 Matrix Integrals and Solitons ...... 4 1.1 Random matrix ensembles ...... 4 1.2 Large n–limits ...... 7 1.3 KP hierarchy ...... 9 1.4 Vertex operators, soliton formulas and Fredholm determinants . 11 1.5 Virasoro relations satisfied by the Fredholm determinant ...... 14 1.6 Differential equations for the probability in scaling limits . . . . . 16 2 Recursion Relations for Unitary Integrals ...... 21 2.1 Results concerning unitary integrals ...... 21 2.2 Examples from combinatorics ...... 25 2.3 Bi-orthogonal polynomials on the circle and the Toeplitz lattice 28 2.4 Virasoro constraints and difference relations ...... 30 2.5 Singularity confinement of recursion relations ...... 33 3 Coupled Random Matrices and the 2–Toda Lattice ...... 37 3.1 Main results for coupled random matrices ...... 37 3.2 Link with the 2–Toda hierarchy ...... 39 3.3 L U decomposition of the moment matrix, bi-orthogonal polynomials and 2–Toda wave operators ...... 41 3.4 Bilinear identities and τ-function PDE’s ...... 44 3.5 Virasoro constraints for the τ-functions ...... 47 3.6 Consequences of the Virasoro relations ...... 49 3.7 Final equations ...... 51 4 Dyson Brownian Motion and the Airy Process ...... 53 4.1 Processes ...... 53 4.2 PDE’s and asymptotics for the processes ...... 59 4.3 Proof of the results ...... 62 5 The Pearcey Distribution ...... 70 5.1 GUE with an external source and Brownian motion ...... 70 5.2 MOPS and a Riemann–Hilbert problem ...... 73 VI Contents

5.3 Results concerning universal behavior ...... 75 5.4 3-KP deformation of the random matrix problem ...... 79 5.5 Virasoro constraints for the integrable deformations ...... 84 5.6 A PDE for the Gaussian ensemble with external source and the Pearcey PDE ...... 88 A Hirota Symbol Residue Identity ...... 92 References ...... 93 Integrable Systems, Random Matrices and Random Processes

Mark Adler

Department of Mathematics, Brandeis University, Waltham, MA 02454, USA, [email protected]

Introduction

Random matrix theory, began in the 1950’s, when E. Wigner [58] proposed that the local statistical behavior of scattering resonance levels for neutrons of heavy nucleii could be modeled by the statistical behavior of eigenvalues of a large random matrix, provided the ensemble had orthogonal, unitary or symplectic invariance. The approach was developed by many others, like Dyson [30, 31], Gaudin [34] and Mehta, as documented in Mehta’s [44] famous treatise. The field experienced a revival in the 1980’s due to the work of M. Jimbo, T. Miwa, Y. Mori, and M. Sato [36, 37], showing the Fredholm determinant involving the sine kernel, which had appeared in random matrix theory for large matrices, satisfied the fifth Painlevétranscendent; thus linking random matrix theory to integrable mathematics. Tracy and Widom soon applied their ideas, using more efficient function-theoretic methods, to the largest eigenvalues of unitary, orthogonal and symplectic matrices in the limit of large matrices, with suitable rescaling. This lead to the Tracy–Widom distributions for the 3 cases and the modern revival of random matrix theory (RMT) was off and running. This article will focus on integrable techniques in RMT, applying Virasoro gauge transformations and integrable equation (like the KP) techniques for finding Painlevé– like ODE’s or PDE’s for probabilities that are expressible as Fredholm determinants coming up in random matrix theory and random processes, both for finite and large n–limit cases. The basic idea is simple – just deform the probability of interest by some time parameters, so that, at least as a function of these new time parameters, it satisfies some integrable equations. Since in RMT you are usually dealing with matrix integrals, roughly speaking, it is usually fairly obvious which parameters to “turn on,” although it always requires an argument to show you have produced “τ–functions” of an integrable system. Fortunately, to show a system is integrable, you ultimately only have to check bilinear identities and we shall present very general methods 2 Mark Adler to accomplish this. Indeed, the bilinear identities are the actual source of a sequence of integrable PDE’s for the τ-functions. Secondly, because we are dealing with matrix integrals, we may change coordinates without changing the value of the integral (gauge invariance), leading to the matrix integrals being annihilated by partial differential operators in the artificially introduced time and the basic parameters of the problem – so-called Virasoro identities. Indeed, because the most useful coordinate changes are often “S1–like” and because the tangent space of DiffpS1q at the identity is the Virasoro Lie algebra (see [41]), the family of annihilating operators tends to be a subalgebra of the Virasoro Lie algebra. Integrable systems possess vertex algebras which infinitesimally deform them and the Virasoro algebras, as they explicitly appear, turn out to be generated by these vertex algebras. Thus while other gauge transformation are very useful in RMT, the Virasoro generating ones tend to mesh well with the integrable systems. Fi- nally, the PDE’s of integrable systems, upon feeding in the Virasoro relations, lead, upon setting the artificially introduced times to zero, to Painlevé-like ODE or PDE for the matrix integrables and hence for the probabilities, but in the original parameters of the problem! Sometimes we may have to introduce “extra parameters,” so that the Virasoro relations close up, which we then have to eliminate by some simple means, like compatibility of mixed partial derivatives. In RMT, one is particularly interested in large n (scaled) limits, i.e. central limit type results, usually called universality results; moreover, one is interested in getting Painlevétype relations for the probabilities in these limiting relations, which amounts to getting Painlevétype ODE’s or PDE’s for Fredholm determinants involving limiting kernels, analogous to the sine kernel previously mentioned. These relations are analogous to the heat equation for the Gaussian kernel in the central limit theorem (CLT), certainly a re- vealing relation. There are two obvious ways to derive such a relation; either, get a relation at each finite step for a particularly “computable or integrable” sequence of distributions (like the binomial) approaching the Gaussian, via the CLT, and then take a limit of the relation, or directly derive the heat equation for the actual limiting distribution. In RMT, the same can be said, and the integrable system step and Virasoro step mentioned previously are thus applied directly to the “integrable” finite approximations of the limiting case, which just involve matrix integrals. After deriving an equation at the finite n–step, we must ensure that estimates justify passage to the limit, which ultimately involves estimates of the convergence of the kernel of a Fredholm determinant. If instead we decide to directly work with the limiting case without passing through a limit, it is more subtle to add time parameters to get integrability and to derive Virasoro relations, as we do not have the crutch of finite matrix integrals. Nonetheless, working with the limiting case is usually quite insightful, and we include one such example in this article to illustrate how the limiting cases in RMT faithfully remember their finite – n matrix integral ancestry. Integrable Systems, Random Matrices and Random Processes 3

In Section 1 we discuss random matrix ensembles and limiting distributions and how they directly link up with KP theory and its vertex operator, leading to PDE’s for the limiting distribution. This is the only case where we work only with the limiting distribution. In Section 2 we derive recursion relations in n for n-unitary integrals which include many combinatoric generating functions. The methods involve bi-orthogonal polynomials on the circle and we need to introduce the integrable “Toeplitz Lattice,” an invariant subsystem of the semi-infinite 2–Toda lattice of Uéno–Takasaki [55]. In Section 3 we study the coupled random matrix problem, involving bi-orthogonal polynomials for a nonsymmetric R2 measure, and this system involves the 2–Toda lattice, which leads to a beautiful PDE for the joint statistics of the ensemble. In Section 4 we study Dyson Brownian motion and 2 associated limiting processes – the Airy and Sine processes gotten by edge and bulk scaling. Using the PDE of Section 3, we derive PDE’s for the Dyson process and then the 2 limiting processes, by taking a limit of the Dyson case, and then we derive for the Airy process asymptotic long time results. In Section 5 we study the Pearcey process, a limiting process gotten from the GUE with an external source or alternately described by the large n behavior of 2n-non-intersecting Brownian? motions starting at x 0 at time t 0, with? n conditioned to go to n, and the other n conditioned to go to n at t 1, and we 1 observe how the motions diverge at t 2 at x 0, with a microscope of resolving power n1{4. The integrable system involved in the finite n problem is the 3-KP system and now instead of bi-orthogonal polynomials, multiple orthogonal polynomials (MOPS) are involved. We connect the 3-KP system and the associated Riemann–Hilbert (RH) problem and the MOP’s. The philosophy in writing this article, which is based on five lectures de- livered at CRM, is to keep as much as possible the immediacy and flow of the lecture format through minimal editing and so in particular, the five sections can read in any order. It should be mentioned that although the first section introduces the basic structure of RMT and the KP equation, it in fact deals with the most sophisticated example. The next point was to pick five examples which maximized the diversity of techniques, both in applying Vira- soro relations and using integrable systems. Indeed, this article provides a fair but sketchy introduction to integrable systems, although in point of fact, the only ones used in this particular article are invariant subsystems (reductions) and lattices generated from the n-KP, for n 1, 2, 3. The point being that a lot of the skill is involved in picking precisely the right integrable system to deploy. Fortunately, if some sort of orthogonal polynomials are involved, this amounts to deforming the measure(s) intelligently. For further reading see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. 4 Mark Adler Acknowledgement

These lectures present joint work, in the fullest sense, with Pierre van Moer- beke, Pol Vanhaecke and Taka Shiota, who I must thank for all the fun we had and also for enduring my teasing. I also wish to thank Alexei Borodin, Persi Diaconis, Alexander Its, Kurt Johansson, Ken McLaughlin, Craig Tracy, Harold Widom, and Nick Witte for many helpful conversations and insightful remarks – the kind of remarks which make more sense as time goes on. Let me also thank Pol Vanhaecke for reading the manuscript and making helpful suggestions, in addition to producing the various ﬁgures that appear. Finally, I wish to thank Louise Letendre of CRM for her beautiful typing job of a messy and hard to read manuscript. The support of a National Science Foundation grant #DMS-04-06287 is gratefully acknowledged.

1 Matrix Integrals and Solitons

1.1 Random matrix ensembles

Consider the probability ensemble over n n Hermitian matrices ³ tr V pMq p qP e dM P M P HpEq ³sp M E tr V pMq p qP e dM ³sp M R ± n p q ∆2 pzq e V zi dz ³En n ±1 i n , (1) 2 p q V pziq Rn ∆n z 1 e dzi ± p q p q p i1q p q with ∆ z i¤i, zi zj det zj i¤i, the Vandermonde and with V x j¤n j¤n a “nice” function so the integral makes sense, sppMq means the spectrum of M and E R,

HnpEq tM an Hermitian n n matrix | sppMq Eu , ¹n ¹ ¹ dM dMii d Real Mij d Imag pMij q , i1 i j i j the induced Haar measure on n n Hermitian matrices, viewed as T SLpn, Cq{ SUpnq , I and where we have used the Weyl integration formula [35] for

1 M U diagpz1, . . . , znqU , ¹n 2 p q dM ∆n z1, . . . , zn dzi dU . 1 Integrable Systems, Random Matrices and Random Processes 5

In order to recast P pMq so that we may take a limit for large n and avoid 8-fold integrals, we follow the reproducing method of Dyson [30, 31]. Let pkpzq be the monic orthogonal polynomials: » V pzq pipzqpjpzqe dz hiδij (2) R and remember pdet Aq2 detpAAT q . (3) Compute » ¹n p q 2 p q V zi ∆n z e dzi Rn » 1 ¹n V pziq det pi1pzjq 1¤i, det pk1pz`q 1¤k, e dzi n ¤ ¤ R j n ` n 1 n » ¸ 1 ¹ ππ V pzkq p1q pπpkq1pzkqpπ1pkq1pzkqe dzk 1 R π,π PSn 1 » n¹1 2 p q V pzq n! pk z e dz (orthogonality) 0 R n¹1 n! hk , (4) 0 and so using (3) and (4), conclude P M P H pEq » n ¸n ¹n 1 p q ± V zi n det pj1pzkqpj1pz`q e dzi n! h n 1 i 1 E 1 1¤k, 1 ¤ » ` n 1 ¹n det Knpzk, z`q 1¤k, dzi , n! n ¤ E ` n 1 with the Christoﬀel–Darboux kernel ¸n hn pϕnpyqϕn1pzq ϕnpzqϕn1pyqq Knpy, zq : ϕjpyqϕjpzq , (5) h y z j1 n 1 and a V pxq{2 ϕjpxq e pj1pxq{ hj1 . Thus by (2) »

ϕipxqϕjpxq dx δij , R and so we have 6 Mark Adler »

Knpy, zqKnpz, uq dz Knpy, uq , »R (6) Knpz, zq dz n , R – the reproducing property, which yields the crucial property: » det Knpzi, zjq 1¤i, dzm1, . . . , dzn Rnm j¤n pn mq! det Knpzi, zjq 1¤i, . (7) j¤m ± n Ñ k nk Replacing E 1 dziR in (1) and integrating out zk1, . . . , zn via the producing property yields:

“P (one eigenvalue in each rzi, zi dzis, i 1, . . . , k) ¹k 1 p q n det Kn zi, zj 1¤i, dzi , ” (8) ¤ k j k 1 heuristically speaking. Setting ¤ E dzi YEi ,

dziPE and using Poincar´e’sformula ¸ ¸ ¸ P pYEiq P pEiq P pEi X Ejq P pEi X Ej X Ekq , i i j i j k yields the Fredholm determinant1 » ¸8 ¹k P p cq p qk Ep q P M Hn E 1 λ det Kn zi, zj 1¤i, dzi ¤¤ ¤ k1 z1 zk j k 1 λ1 p Eq det I λKn (9) λ1 with kernel Ep q p q p q Kn y, z Kn y, z IE z and with IEpzq the indicator function of the set E, and more generally see [48], p qk B k P q 1 p Eq P (exactly k-eigenvalues E B det I λKn . (10) k! λ λ1

1 Ec is the complement of E in R. Integrable Systems, Random Matrices and Random Processes 7

1.2 Large n–limits

The Fredholm determinant formulas (9) and (10), enable one to take large n– limits of the probabilities by taking large n–limits of the kernels Knpy, zq. The ﬁrst and most famous such law is for the Gaussian case V pxq x2, although it has been extended far beyond the Gaussian case [38, 29].

Wigner’s semi-circle law:

The density of eigenvalues converges (see Fig. 1) in the sense of measure: $ ? ? & 1 2n z dz, |z| ¤ 2n , p q Ñ π Kn z, z dz % ? (11) 0, |z| ¡ 2n

and so »

Expp# eigenvalues in Eq Knpz, zq dz. E This is a sort of Law of Large Numbers. Is there more reﬁned universal behavior, a sort of Central Limit Theorem? The answer is as follows:

Bulk scaling limit: ? ? The density of eigenvalues near z 0 is 2n{π and so π{ 2n average distance between eigenvalues. Magnifying at z 0 with this rescaling πx πy yπ sin πpx yq lim Kn ? , ? d ? dy nÑ8 2n 2n 2n πpx yq

: Ksinpx, yq dy , (12) with

Fig. 1. 8 Mark Adler π lim P exactly k eigenvalues P ? r2a, 2as nÑ8 2n k k p1q B r2a,2as detpI λK q . (13) B sin k! λ λ1

Moreover, » πa p q p r2a,2asq f x, λ det I λKsin exp dx 0 x with d pxf 2q2 4pxf 1 fqpf 12 xf 1 fq 0 1 , (14) dx and boundary condition: λx λ 2 fpx, λq x2 , x 0 . π π

The O.D.E. (14) is Painlev´eV, and this is the celebrated result of Jimbo, Miwa, Mori, and Sato [36, 37].

Edge scaling limit: ? The? density of eigenvalues near the edge, z 2n of the? Wigner semi-circle is 2n1{6. Magnifying at the edge with the rescaling 1{ 2n1{6: ? u ? v ? v lim Kn 2n ? , 2n ? d 2n ? nÑ8 2n1{6 2n1{6 2n1{6

: KAirypu, vq dv , (15) with » 8

KAirypu, vq Aipx uqAipx vq dx , »0 (16) 1 8 u3 Aipuq cos xu dv . π 0 3 ? ? 1{6 Setting λmax 2n u{ 2n ? ? 1{6 lim P p 2n pλmax 2nq ¤ uq nÑ8 » 8 p ru,8sq p q 2p q det I KAiry exp α u g α dα u – the Tracy–Widom distribution , (17) with g2 xg 2g3 (18) Integrable Systems, Random Matrices and Random Processes 9 and boundary condition:

expp2{3x3{2q gpxq ? , x Ñ 8 . (19) 2 πx1{4

Equation (18) is Painlev´eII and (17) is due to Tracy–Widom [49].

Hard edge scaling limit:

Consider the Laguerre ensemble of n n Hermitian matrices:

V pzq ν{2 z{2 e dz z e Ip0,8qpzq dz . (20)

Note z 0 is called the hard edge, while z 8 is called the soft edge. The density of eigenvalues for large n has a limiting shape and the density of eigenvalues near z 0 is 4n. Rescaling the kernel with this magniﬁcation: » 1 ? ? pνq u v v p q 1 p q p q lim Kn , d : Kν u, v dv sJν s u Jν s v ds dv nÑ8 4n 4n 4n 2 ? ? ? 0 ? ? ? J p uq vJ 1 p vq J p vq uJ 1 p uq ν ν ν ν dv (21) 2pu vq yields the Bessel kernel, while one ﬁnds: 1 lim P no eigenvalue P r0, xs nÑ8 4n » x p q p r0,xsq f v det I Kν exp du , (22) 0 u with pxf 2q2 4pxf 1 fqf 12 px ν2qf 1 f f 1 0 , (23) and boundary condition: 1 fpxq c x1ν 1 x , c r22ν2Γ p1 νqΓ p2 νqs1 . ν 2p2 νq ν

Equation (23) is Painlev´eV and is due to Tracy–Widom [50].

1.3 KP hierarchy

We give a quick discussion of the KP hierarchy. A more detailed discussion can be found in [28, 56]. Let L Lpx, tq be a pseudo-diﬀerential operator and Ψ px, t, zq its eigenfunction (wave function). The KP hierarchy is an isospectral deformation of L:2

2 T Ψ is the eigenfunction of L adjoint:L , pAq :differential operator part of A. 10 Mark Adler B L n 1 rpL q,Ls,L D a px, tqD , n 1, 2,..., Bt x 1 x n (24) B D , t pt , t ,... q x Bx 1 2 with Ψ parametrized by Sato’s τ-function and satisfying

°8 p r 1sq pxz t ziq τ t z Ψ px, t, zq e 1 i τptq ° pxz 8 t ziq e 1 i 1 0p1{zq , z Ñ 8 (25) and B Ψ n zΨ LΨ , pL qΨ , Bt n (26) B T Ψ p T qn zΨ L Ψ , L Ψ , Btn with x2 x3 rxs : x, , ,... . 2 3 Consequently the τ-function satisﬁes the crucial formal residue identity.

Bilinear identity for τ-function: ¾ ° 8pt t1 qzi 1 1 1 1 8 e 1 i i τpt rz sqτpt rz sq dz 0, @t, t P C , (27) 8 which characterizes the KP τ-function. This is equivalent (see Appendix for proof) to the following generating function of Hirota relations (a pa1, a2,... q arbitrary):

8 ¸ °8 r a`B{Bt` sjp2aqsj1pBtqe ` 1 τptq τptq 0 , (28) j0 where r B 1 B 1 B B B Bt , , ,... , Bt , ,... , (29) Bt1 2 Bt2 3 Bt3 Bt1 Bt2 and 8 °8 ¸ tizi j e 1 : sjptqz (sjptq elementary Schur polynomials) (30) j0 and B B pB q p q p q p q p q p t f t g t : p B , B ,... f t y g t y (Hirota symbol). (31) y y2 y0 Integrable Systems, Random Matrices and Random Processes 11

This yields the KP hierarchy B4 B 2 B B 3 4 τptq τptq 0 (32) Bt1 Bt2 Bt1 Bt3 . . equivalent to B 4 B 2 B2 B2 2 3 4 log τ 6 log τ 0 B B B B B 2 t1 t2 t1 t3 t1 (KP equation). (33) . . The p-reduced KP corresponds to the reduction: p p p pq L Dx diﬀ. oper. L and so B L pn pn rpL q,Ls rL ,Ls 0 . (34) Btpn p 2: KdV

τ τpt1, t3, t5,... q (ignore t2, t4,... ) (35) Ψ px, t, zq Ψpx, t, zq . (36)

1.4 Vertex operators, soliton formulas and Fredholm determinants Vertex operators in integrable systems generate the tangent space of solutions and Darboux transformations; in other words, they yield the deformation theory. We now present KP – vertex operator:3 ° ° 1 8pziyiqt 8pyiziq1{i B{Bt Xpt, y, zq e 1 i e 1 i , (37) z y and the “τ-space” near τ parametrized: τ εXpt, y, zqτ. Moreover τ εXpt, y, zqτ is a τ-function, not just inﬁnitesimally, so it satisﬁes the bilinear identity. This vertex operator was used in [28] to generate solitons, but it also plays a role in generating Kac–Moody Lie algebras [40]. The identities that follow in Sections 1.4, 1.5, 1.6 were derived by Adler–Shiota–van Moerbeke in [16], carefully and in great detail. ° 8p i iq 3 Xpt, y, zqfptq 1{pz yq e 1 z y ti fpt ry 1s rz 1sq, z y and z, y are large complex parameters, and how we expand the operator X shall depend on the situation. 12 Mark Adler

Link with kernels:

We have the diﬀerential Fay Identity (Adler–van Moerbeke [1]) 1 Xpt, y, zqτptq D1 Ψ px, t, yqΨ px, t, zq , (38) τptq x

1 where since Dx is integration, the r.h.s. of (38) has the same structure as the Airy and Bessel kernels of (16) and (21). If |yi|, |zi| |yi1|, |zi1|, 1 ¤ i ¤ n 1, then we have the Fay Identity: 1 det D1 Ψ px, t, y qΨ px, t, z q det Xpt, y , z qτptq x i j 1¤i, p q i j ¤ ¤ τ t 1 i, j n j¤n 1 Xpt, y , z q Xpt, y , z qτ . (39) τ 1 1 n n We also have the following

Vertex identities:

Xpy, zqXpy, zq 0 and so eaXpy,zq 1 aXpy, zq , (40) and rXpα, βq,Xpλ, µqs 0 if α µ, β λ . (41) In addition we have the expansion of the vertex operator along the diagonal:

¸8 k ¸8 1 pz yq p q Xpt, y, zq y`kW k , z y k! ` k0 `8 B p1q p q p p0q q Heisenberg: Wn : n tn Wn δn0 , (42) Btn ¸ B ¸ B2 ¸ p2q p qp q Virasoro: Wn : 2 iti B B B iti jtj ¥ tin ti tj i,i n 1 i j n i j n B pn 1q pnqtn Btn and from the commutation relations: rXpα, βq,Xpλ, µqs nT pα, β, λq npα, β, µq Xpλ, µq , npλ, µ, zq : δpλ, zqepµλq B{Bz , 8 (43) 1 ¸ z ` δpλ, zq : , z 8 λ conclude the vertex Lax equation: Integrable Systems, Random Matrices and Random Processes 13 B 1 p q z`1Y pzq W 2 ,Y pzq (44) Bz 2 ` with 1 1 4 Y pzq Xpt, ωz, ω zq, ω, ω P ξp , ` ¥ p and p{` , (45) or a linear combination of such Xpt, ωz, ω1zq.

A Fredholm determinant style soliton formula:

A classical KP soliton formula [28] is as follows: ¹n ° 1 a 8 k k aiXpyi,ziq j pz y qtk τptq e τ0 det δij e k 1 j i . τ0 zj yi 1¤i, 1 τ 1 ¤ ordered 0 j n

We now relax the condition that τ0 1 and setting |yi|, |zi| |yi1|, |zi1|, 1 ¤ i ¤ n 1, compute using the Fay identity (39) and diﬀerential Fay (38):

° τ˜ptq 1 n p q 1 aiX yi,zi p q p q : p qe τ t τ t τ t ¹n 1 p q eaiX yi,zi τptq τptq 1 ordered 1 ¹n 1 a Xpy , z q τptq τptq i i i 1 ordered ¸ ¸ 1 p q p q p q p q τ aiX yi, zi τ aiajX yi, zi X yj, zj τ τ t ¤ ¤ ¤ ¤ 1 i n 1 i j n ¹n ¹n ai Xpyi, ziq τ 1 1 ordered p q p q ¸ ¸ X yi,zi τ X yi,zj τ Xpy , z qτ ai , aj 1 a i i det τ τ i p q p q τ X yj ,zi τ X yj ,zj τ 1¤i¤n 1¤i j¤n a , a i τ j τ Xpyi, zjqτ det aj τ 1¤i, j¤n Xpyi, zjqτ det δij aj “Fredholm expansion” of determinant τ 1¤i, j¤n

4 ξp the pth roots of unity. 14 Mark Adler 1 p q p q det δij ajDx Ψ x, t, yi Ψ x, t, yj 1¤i, . j¤n 1 Replacing yi Ñ ωzi, zi Ñ ω zi, ai λδz b a z a pi 1qδz, δz , n Ñ 8 i n 1 yields the continuous (via the Riemann Integral).

Soliton Fredholm determinant: ³ p q 1 τ t, E 1 λ Xpt,ωz,ω zq dz E : e E τptq detpI λk q (46) τptq τptq with kernel Ep q 1 p q p 1 q p q r s k y, z Dx Ψ t, ωy Ψ t, ω z IE z ,E a, b . More generally, for p-reduced K-P, replace in (46) ¸ 1 Xpt, y, zq Ñ Y pt, y, zq : aωbω1 Xpt, ωy, ω zq , (47) 1P ¸ ω,ω ¸ξp 1 Ep q Ñ 1 p q 1 p q p q k y, z Dx aωΨ x, t, ωy bω Ψ x, t, ω z IE z (48) 1 ωPξp ω Pξp with ¸ a b ω ω 0 (so Y pt, z, zq is pole free) (49) ω ωPξp and

Xpt, ωz, ω1zq Ñ Y pt, z, zq : Y pzq , (50) ¤k E Ñ ra2i1, a2is . (51) i1

1.5 Virasoro relations satisﬁed by the Fredholm determinant

It is a marvelous fact that the soliton Fredholm determinant satisﬁes a Vira- soro relation as a consequence of the vertex Lax-equation [16]; indeed, compute » b B 1 p q 0 z`1Y pzq W 2 ,Y pzq dz Bz 2 ` a » b 1 p q b`1Y pbq a`1Y paq W 2 , Y pzq dz 2 ` a» b B B 1 p q b`1 a`1 W 2 , Y pzq dz B B ` b a 2 a : δpUq , Integrable Systems, Random Matrices and Random Processes 15 with δ a derivation and hence

δeλU 0 , or spelled out B B ³ `1 `1 1 p2q λ b Y pzq dz b a W , e a 0 . (52) Bb Ba 2 `

Let τ be a vacuum vector for p-KP:

p2q W` τ c`τ , ` kp , k 1, 0, 1,.... (53) Remembering (46) with Y pzq given by (50), and with τptq taken as a vacuum vector, yields p q ³ τ t, E 1 λ Y pzq dz E e E τ detpI λk q , (54) τptq τ and setting ` kp, k 1, 0, 1,..., compute using, (52), (53) and (54): B B ³ ³ `1 `1 1 p2q λ b Y pzq dz λ b Y pzq dz 1 p2q 0 b a W e a τ e a W τ B B ` ` b a 2 2 B B ³ `1 `1 1 p2q λ b Y pzq dz b a pW c q e a τ B B ` ` b a 2 B B 1 p q b`1 a`1 pW 2 c q τptq detpI λkEq . (55) Bb Ba 2 ` `

More generally: setting

¤k 1{p ra, bs Ñ E : ra2i1, a2is , i1 B B ¸2k B b`1 a`1 Ñ a`1 : B paq , Bb Ba j Ba ` j1 j deduce 1 p q 1{p B paq pW 2 c q τptq detpI λkE q 0 . kp 2 kp kp with p2q p q p q ¥ Wkp τ t ckpτ t , k 1 . (56) Since changing integration variables in a Fredholm determinant leaves the determinant invariant, change variables:

Ñ p Ñ p z z λ, ai ai Ai , and 16 Mark Adler

¤k 1{p E Ñ E rA2i1,A2is , i1 and E1{p 1{p 11{p 1{p k pλ , λ q kE pz, z1q Ñ KEpλ, λ1q : I pλ1q , (57) pλpp1q{2pλ1pp1q{2p E yielding

1{p detpI µKEq detpI µkE q ³ p q 1 µ { Y pzq dz τ t, E e E1 p , (58) τ τptq and so (56) yields the p reduced Virasoro relation: p q 1 p p2q q p q p Eq Bk A 2 Wkp ckp τ t det I µK 0 , (59) with p2q p q p q ¥ Wkp τ t ckpτ t , k 1 .

1.6 Diﬀerential equations for the probability in scaling limits

Now we shall derive diﬀerential equations for the limiting probabilities discussed in Section 1.2 using the integrable tools previously developed.

Airy edge limit:

Remembering the edge limit for Hermitian eigenvalues of (15) and (16): ? ? p 1{16p q P cq p E q lim P 2n λmax 2n E det I KAiry (60) nÑ8 with » 8

KAirypu, vq Aipx uqAipx vq dx , »0 (61) 1 8 u3 Aipuq cos xu du . π 0 3 Consider the KdV reduction t pt1, 0, t3, 0, t5,... q p 2, (62) t0 p0, 0, 2{3, 0, 0,... q with initial conditions: $ ? p q p p q2q ' Ψ x, t0, z 2 πzA x z & 2 exz2z {3 1 0p1{zq , ' (63) %' p 2 q p q 2 p q Dx x Ψ x, t0, z z Ψ x, t0, z . Integrable Systems, Random Matrices and Random Processes 17

Under the KP (KdV) ﬂow: $ ' 2B2 'x qpx, t q Ñ qpx, tq `n τptq , ' 0 B 2 & t1 ° 1 8 τpt rz sq (64) ' p q Ñ p q xz 1 tizi 'Ψ x, t0 Ψ x, t e p q , %' τ t τpt0q Ñ τptq . where τptq turns out to be the well-known Konsevich integral [16, 18], satisfying a vacuum condition as a consequence of Grassmannian invariance, to wit:

Kontsevich integral: $ ³ p 3{ 2 q ' dXe Tr X 3 X Z & p q HN³ τ t limNÑ8 p 2 q , dXe Tr X Z (65) ' HN %' 1 2 Z diag: t Tr Z n δ . n n 3 n,3 Vacuum condition:

p2q 1 ¥ W2k τ 4 δk0 τ, k 1 . (66) Grassmannian invariance condition: $ " * j ' B 'W : span Ψpx, t0, zq , j 0, 1, 2,... , ' C Bx &' x0 'z2W W pKdV q , AW W, (67) ' ' ' 1 B 1 %A 2z2 ,A2Ψp0, t , zq z2Ψp0, t , zq . 2z Bz 4z2 0 0 We have the initial kernel: » I pλ1q 8 ? ? KEpλ, λ1q E Ψpx, t , λqΨpx, t , λ1q dx t0 1{4 11{4 0 0 2λ λ 0 » 8 1 1 2πIEpλ q Aipx λqAipx λ q dx 0 Ñ ÝÝÝÑt0 t Ep 1q Kt λ, λ . (68)

Conditions on τpt, Eq: 1 τpt, Eq τptq det I KE (by (58)) 2π t p q p E q τ t0 det I KAiry at t t0 (by (68)) , (69) 18 Mark Adler which satisﬁes (33) and (35): B 4 B2 B2 2 4 log τ 6 log τ 0 (KdV) , (70) B B B B 2 t1 t1 t3 t1 and by (59), (42) and (62) we ﬁnd for τpt, Eq

Virasoro constraints: B ¸ B 2 1 t1 B1pAqτ iti τ , Bt 2 Bt 4 1 i¥3 i 2 (71) B 1 ¸ B 1 B pAqτ it τ . 0 Bt 2 i Bt 16 3 i¥1 i

Replace t-derivatives of τpt, Eq at t0 with A derivatives in KdV:

B B2 B2 B τ 2 1 B τ ,B τ τ, . . . , B B τ τ, . . . , 1 B 1 B 2 1 0 B B B t1 t1 t1 t3 2 t1 at t t0 , (72) yielding

Theorem 1.1 (Adler–Shiota–van Moerbeke [16]). ? ? p 1{6p q P cq p E q R : B1 log lim P 2n λmax 2n E B1 log det I KAiry nÑ8

τpt0,Eq B1 log τpt0q

B1 log τpt0,Eq satisﬁes 3 p 1 q p q2 B1 4 B0 2 R 6 B1R 0 . (73) Setting E pa, 8q yields:

R3 4aR1 2R 6R12 0 . (74) Setting R g12 ag2 g4,R1 g2 yields g2 2g3 ag (Painlev´eII) . (75) Integrable Systems, Random Matrices and Random Processes 19

Hard edge limit:

Remembering the hard edge limit (21) for the Hermitian Laguerre ensemble (20): P 1 p Eq lim P no eigenvalues E det I Kν , nÑ8 4n » (76) 1 1 ? ? Kν pu, vq sJν ps uqJν ps vq ds 2 0 deﬁned in terms of Bessel functions, consider the KdV reduction with initial conditions: $ &'Ψpx, 0, zq exzB p1 xqz exz 1 0p1{zq , ' pν2 1 q (77) % D2 4 Ψpx, 0, zq z2Ψpx, 0, zq x px 1q2 with (see [19]) » ? ez2ν1{2 8 zν1{2euz Bpzq : ε zH pizq du , ν p 1 q p 2 qν1{2 Γ ν 2 1 u 1 where c π 1 1 ε i eiπν{2, ν . 2 2 2 Under the KdV ﬂow: ν2 1 4 ,Ψpx, 0, zq, τp0q Ñt qpx, tq,Ψpx, t, zq, τptq , (78) x2 1 where τptq is both a Laplace matrix integral and a vacuum vector [16, 18] due to Grassmannian invariance:

Laplace integral: ³ ³ ν1{2 Tr Z2X Tr XY 2 p q H dX det X e H dY S0 Y e τptq lim S ptq N ³ N 1 Tr X2Z NÑ8 dX e HN { n X with Z diag: tn 1 n tr Z , Hn Hn (matrices with non-negative spectrum) and S1ptq, S0pY q are symmetric functions,

Vacuum condition: p2q p q2 ¥ W2k τ 2ν 1 τδk0, k 1 , (79) 20 Mark Adler

Grassmannian invariance: $ 2 & z W W, AW W, % 1 B 2 2 1 p q 2 p q A 2 z Bz 1 , 4A 2A ν 4 Ψ 0, 0, z z Ψ 0, 0, z . We have the initial kernel:

E p 1q Kpx,tq λ, λ p 1q ? ? IE λ 1 p 1q { 1 { Dx Y x, t, λ, λ (see (48), (49) and (57)) 2λ1 4λ 1 4 p 1q ¸ ? ¸ ? IE λ 1 1 D aΨ px, t, λq bΨ px, t, λ q 1{4 11{4 x 2λ λ » I pλ1q x ieiπν{2 ? eiπν{2 ? : E ? Ψpx, t, λq ? Ψpx, t, λq 2λ1{4λ11{4 2π 2π 1 eiπν{2 ? ieiπν{2 ? ? Ψpx, t, λ1q ? Ψpx, t, λ1q dx 2π 2π » 1 ? ? 1 1 1 IEpλ q sJν ps λqJν ps λ q ds at px, t0q p1 i, e1q , (80) 2 0 and under the t-ﬂow Ñ KE pλ, λ1q ÝÝÝÑt0 t KE pλ, λ1q . p1i,e1q px,tq Conditions on τpt, Eq:

p q p q p E q p qp Eq p q p q τ t, E τ t det I Kpx,tq τ t0 I Kν at x, t0 1 i, e1 (by (58) and (80)) , (81) satisﬁes (33) and (35) : B 4 B2 B2 2 4 log τ 6 log τ 0 (KdV) , (82) B B B B 2 t1 t1 t3 t1 and by (59), (42), (62) and (81), the

Virasoro constraints: ¸ B ? B 1 1 2 B0pAqτ iti 1 2 ν τ , 2 Bti Bt1 4 i¥1 ¸ B B2 ? B p q 1 1 B1 A τ iti 2 1 τ . 2 Bt 2 Bt Bt i¥1 i 2 1 3 Integrable Systems, Random Matrices and Random Processes 21

Replace t-derivatives of τpt, Eq at p1 i, e1q with A derivatives in KdV: i Bτ 1 1 B2τ i Bτ B pAqτ ν2 τ , B pAq ,... at p1 i, e q 0 B 1 B 2 B 1 2 t1 4 4 t1 2 t3 yielding Theorem 1.2 (Adler–Shiota–van Moerbeke [16]). P E p Eq R : log lim P no eigenvalues log det I Kν nÑ8 4n τpi, 0, 0,... ; Eq log τpi.0, 0,... ; Rq satisﬁes 4 3 p 2q 2 p 1 q p qp 2 q p 2 q2 B0 2B0 1 ν B0 B1 B0 2 R 4 B0R B0 R 6 B0 R 0 . Setting : BR E p0, xq, f x Bx yields

f 2 6f 12 4ff 1 px ν2qf 1 f f 3 0 (Painlev´eV) . x x x2 x2 2x2

2 Recursion Relations for Unitary Integrals

2.1 Results concerning unitary integrals

Many generating functions in a parameter t for combinatorial problems are expressible in the form of unitary integrals Inptq over Upnq (see [20, 22, 47, 51]). Our methods can be used to either get a diﬀerential equation for Inptq in t [2] or a recursion relation in n [3] and in the present case we concentrate on the latter. Borodin ﬁrst got such results [26] using Riemann–Hilbert techniques. Consider? the following basic objects (∆npzq is the Vandermonde determinant, i 1):

Unitary integral: » pεq ε p q In det M ρ M dM Upnq » ¹n 1 | p q|2 ε p q dzk ∆n z zkρ zk n! pS1qn 2πizk » k1 1 1 dz det zεi j ρpzq , (83) ¤ 1 S1 2πiz 1 i , j1¤n 22 Mark Adler with weight ρpzq:

p q p 1q 1 1 2 p q P1 z P2 z γ p qγ1 p qγ2 p 1 1qγ1 ρ z e z 1 d1z 1 d2z 1 d1 z 2 p 1 1qγ2 1 d2 z , (84)

¸N1 i ¸N2 i u z u z P pzq : i ,P pzq : i , (85) 1 i 2 i 1 1 and we introduce the

Basic recursion variables:

I I x : p1qn n , y : p1qn n , (86) n p0q n p0q In In p0q p0q I I v : 1 x y n 1 n 1 , (87) n n n p0q2 In and so n¹1 p0q p p0qqn p qni In I1 1 xiyi , (88) 1 thus p q t p0qu x, y recursively yields In . We also introduce the fundamental semi-inﬁnite matrices:

Toeplitz matrices:

x1y0 1 x1y1 0 0 ... x2y0 x2y1 1 x2y2 0 ... L px, yq : , (89) 1 x3y0 x3y1 x3y2 1 x3y3 ...... T p q L2 : L1 y, x , in terms of which we write the following:

Recursion matrices:5

pnq 2 1 1 1 L : paI bL1 cL qP pL1q cpn γ γ γqL1 , 1 1 1 1 2 (90) pnq p 2q 1p q p 2 2 q L2 : cI bL2 aL2 P2 L2 a n γ1 γ2 γ L2 . There exists the following basic involution :

5 1 I in (90) is the semi-inﬁnite identity matrix and Pi pzq dPipzq{dz. Integrable Systems, Random Matrices and Random Processes 23

Basic involution:

: z Ñ z1, ρpzq Ñ ρpz1q , p0q Ø p0q Ø In In ,In In ,

xn Ø yn, a Ø c, b Ø b, γ Ñ γ Ø T pnq Ø pnqT L1 L2 , L1 L2 . (91) Self-dual case:

p q p 1q ùñ T pnq pnqT ρ z ρ z , xn yn L1 L2 , L1 L2 . (92) Let us deﬁne the “total discrete derivative”:6

Bnfpnq fpn 1q fpnq . (93)

We now state the main theorems of Adler–van Moerbeke [3]:

Rational relations for px, yq:

Theorem 2.1. The pxk, ykqk¥1 satisfy 2 ﬁnite step relations:

Case 1.

In the generic situation, to wit: | 1 | | 2| | 1 | | 2| p q d1, d2, d1 d2, γ1 γ1 , γ2 γ2 0 in ρ z , (94) we ﬁnd for n ¥ 1 that $ & pnq pnq BnpL L qn,n pcL1 aL2qn,n 0 pq 1 2 %B p pnq pnqq p 2 q n vnL1 L2 n1,n cL1 bL1 n1,n1 (same)n0 , with 1 pa, b, cq ? p1, d1, d2, d1d2q , (95) d1d2 and the dual equations pq˜ , also hold.

Case 2.

Upon rescaling, γ γ1 P pzqP pz1q ρpzq z p1 zq 1 e 1 2 , (96) and then pq, pq˜ are satisﬁed with pa, b, cq p1, 1, 0q or p0, 1, 1q.

6 The total derivative means you must take account, in writing fpnq, of all the places n appears either implicitly or explicitly, so fpnq gpn, n, n, . . . q in reality. 24 Mark Adler

Case 3.

Upon rescaling, p q p 1q ρpzq zγ eP1 z P2 z , (97) and then pq, pq˜ are satisﬁed for a, b, c arbitrary and in addition ﬁner relations hold: r Γnpx, yq 0 , Γnpx, yq 0 , L P 1pL q L P 1pL q (98) v 1 1 1 n1,n1 2 2 2 n,n p q n Γn x, y : 1 1 nxn . yn p q p q P1 L1 n1,n P2 L2 n ,n 1

If |N1 N2| ¤ 1, where Ni degree Pi, the rational relations of Theo- rem 2.1 become, upon setting zn : pxn, ynq:

Rational recursion relations:

Theorem 2.2. For N1 N2 1 or N1 N2, the rational relations of Theo- rem 2.1 become recursion relations as follows:

Case 1.

Yields inductive rational N1 N2 4 step relations: p q zn Fn zn1, zn2, . . . , znN1N23 .

Case 2.

Yields inductive rational N1 N2 3 step relations: p q zn Fn zn1, zn2, . . . , znN1N22 , with either

N1 N2 or N2 1: pa, b, cq p1, 1, 0q , or

N1 N2 or N2 1: pa, b, cq p0, 1, 1q .

Case 3.

Yields inductive rational N1 N2 1 step relations: p q zn Fn zn1, zn2, . . . , znN1N2 , r upon using Γn and Γn and in the case of the Integrable Systems, Random Matrices and Random Processes 25

Self dual weight: ° N1 u pziziq{i ρpzq e 1 i . One ﬁnds recursion relations:

xn Fnpxn1, xn2, . . . , xn2N q .

2.2 Examples from combinatorics In this section, we give some well-known examples from combinatorics in the notation of the previous section. In the case of a permutation πk, of k– numbers, Lpπkq shall denote the length of the largest increasing subsequence pwq of πk. If πk is only a word of size k from an alphabet of α numbers, L pπkq shall denote the length of the largest weakly increasing subsequence in the word πk. We will also consider odd permutations π on pk, . . . , 1, 1, . . . , kq or pk, . . . , 1, 0, 1, . . . , kq, which means πpjq πpjq, for all j.

1 Example 1. ρpzq etpzz q (self-dual case) » ¸8 2k p0qp q t p P q | p q ¤ q t TrpMM 1q In t : P πk Sk L πk n e dM , k! p q k0 U n the latter equality due to I. Gessel, with ³ p q t TrpMM 1q p q det M e dM x ptq p1qn U n³ , (as in (86)) n t TrpMM 1q Upnq e dM and n¹1 p0qp q p0qp q n p 2qni In t I1 t 1 xi . i1 One ﬁnds a

3-step recursion relation for xi:

p 2 q 1 xn 0 nxn ptpL1qn1,n1 tpL1qnnq (as in (98)) xn p 2 qp q nxn t 1 xn xn1 xn1 (Borodin [26]) which possesses an

Invariant : Φpxn1, xnq Φpxn, xn1q , n Φpy, zq p1 y2qp1 z2q yz (McMillan [43]) . t The initial conditions in the recursion relation are as follows: 1 d p q x 1, x log I p2tq,I 0 I p2tq , 0 1 2 dt 0 1 0 with I0 the hyperbolic Bessel function of the ﬁrst kind (see [19]). 26 Mark Adler

1 2 2 Example 2. ρpzq etpzz qspz z q (self-dual case) Set

odd t P p q u S2k π2k S2k acts on k, . . . , 1, 1, . . . , k oddly , odd t P p q u S2k1 π2k1 S2k1 acts on k, . . . , 1, 0, 1, . . . , k oddly and then one ﬁnds: ? ¸8 » p q2k 2s p P odd | p q ¤ q s TrpM 2M 2q P π2k S2k L π2k n e dM , k! Upnq k0 ? ¸8 p q2k 2s odd P pπ P S | Lpπ q ¤ nq k! 2k 1 2k1 2k 1 k0 2 » 1 B 1 2 2 TrptpMM qspM M qq p q B e dM same t, s , 4 t Upnq t0 as observed by M. Rains [47] and Tracy–Widom [51]. Moreover, ³ p q TrptpMM 1qspM 2M 2qq p q det M e dM x pt, xq p1qn U n³ , n TrptpMM 1qspM 2M 2qq Upnq e dM and n¹1 p0qp q p0qp q n p 2qni In t I1 t 1 xi . i1 One ﬁnds a

5-step recursion relation for xi:

2 nxn tvnpxn1 xn1q2svnpxn2vn1 xn2vn1 xnpxn1 xn1q q0 p 2 q vn 1 xn which possesses the Invariant : Φpxn1, xn, xn1, xn2q same , Ñ n n 1 Φpx, y, z, uq : nyz p1 y2qp1 z2q t 2s xpu yq zpu yq , analogous to the McMillan invariant of the previous example.

1 Example 3. ρpzq p1 zqesz (Case 2 of Theorem 2.1) Set

Sk,α twords πk of length k from alphabet of size αu , Integrable Systems, Random Matrices and Random Processes 27 with 8 ¸ pαsqk Ip0qpsq P pπ P S | Lpwqpπ q ¤ nq n k! k k,α k »k0 1 detpI Mqαes tr M dM , Upnq the latter identity observed by Tracy–Widom [52]. Then setting in

Case 2.

P1pzq 0, P2pzq sz, N1 0, N2 1, pa, b, cq p0, 1, 1q,

pnq p q pnq p q L1 n α L1, L2 s I L2 , one ﬁnds the

Recursion relations:

B pn αqL sL pL q n 1 2 nn 1 nn 2 B p q p q p q n n 1 α vn 1L1 sL2 n,n1 L1 L1 n,n same n1

pvn 1 xnynq , with ³ 1 p1qn pdet Mq detpI Mqαes Tr M dM U³pnq xn,yn 1 p qα s tr M Upnq det I M e dM and n¹1 p0qp q p0qp q n p qni In s I1 s 1 xiyi , i1 leading to the

3 and 4 step relations for pxi, yiq:

pn α 1qx y sx y pn α 1qx y sx y 0 , n 1 n n n 1 n n 1 n 1 n v pn α 1qx y s v pn α 2qx y s n n 1 n 1 n 1 n n 2 xnyn1pxnyn1 1q v1 p2 αqx2 s x1px1 1q

px0 y0 1q . 28 Mark Adler

2.3 Bi-orthogonal polynomials on the circle and the Toeplitz lattice It turns out that the appropriate integrable system for our problem is the Toeplitz lattice, an invariant subsystem of the 2–Toda lattice. Indeed xn and yn of Section 2.1 turn out to be dual Hamiltonian variables for the integrable system; moreover xn and yn are nothing but the constant term of the nth bi- orthogonal polynomials on the circle, generated by a natural time deformation of our measure ρpzq of (84). These things are discussed by Adler–van Moerbeke in [2, 3] in detail. Consider the bi-orthogonal polynomials and inner product generated by the following measure on S1: ° dz 8pt zks zkq dz ρpz, t, sq e 1 k k , (99) 2πiz 2πiz with Inner product: ¾ ° dz 1 8pt zisiziq xfpzq, gpzqy : fpzqgpz qe 1 i . (100) 2πiz S1 Bi-orthogonal polynomials: p q x p1q p2qy τn1 t, s pn , pm : δn,mhn, hn , n, m 0, 1,.... (101) τnpt, sq The polynomials are parametrized by 2-Toda τ functions:

k ` τnpt, sq detpxz , z yq0¤k, (Toeplitz determinant) ¤ » ` n 1 ¹n ° 1 8 j j dz 2 ptj z sj z q k |∆npzq| e j 1 k k n! p 1qn 2πizk » S k1 ° 8 i i Tr ptiM siM q e 1 dM, n ¥ 1, τ0 1 , (102) Upnq as follows: n p p1q p2qqp q u p r 1s q p r 1sq pn , pn u; t, s τn t u , s , τn t, s u τnpt, sq ³ ° p 8 i iq p q Tr 1 tiM siM p q det uI M e dM U n ³ ° i , Trp tiMisiM q Upnq e dM ³ ° p 8 i iq p 1q Tr 1 tiM siM p q det uI M e dM U n ³ ° i , Trp tiMisiM q Upnq e dM

rxs px, x2{2, x3{3,... q . (103) Integrable Systems, Random Matrices and Random Processes 29

The constant term of our polynomials yield the Dynamical variables:

p q p1qp q p q p2qp q xn t, x pn 0; t, s , yn t, s pn 0; t, s , (104) τn1τn1 vn 1 xnyn 2 ; (105) τn giving rise to an invariant subsystem of the Toda equations: x1y0 1 x1y1 0 0 ... x2y0 x2y1 1 x2y2 0 ... L px, yq : , (106) 1 x3y0 x3y1 x3y2 1 x3y3 ...... p q T p q { p q L2 x, y : L1 y, x , hn τn1 τn , h diag h0, h1 ,... , (107) L˜ hL h1 , L˜ L , (108) $ 1 1 2 2 'B ˜ ' Li n 7 &' rpL˜ q, L˜ s Bt 1 i pT q n , i 1, 2, n 1, 2,... (109) 'BL˜ %' i rp˜nq ˜ s L2 , Li Bsn with # pAq upper tripAq diagpAq , pAq lower tripAq . We ﬁnd, pT q ô Toeplitz lattice:7 $ p1q p1q 'Bx BH By BH &' n v i , n v i Bt n By Bt n Bx i n i n , i 1, 2,..., (110) 'B B p2q B B p2q %' xn Hi yn Hi vn , vn Bsi Byn Bsi Bxn i ¸8 p q tr L dx ^ dy H j : j , ω k k , v 1 x y , (111) i i v n n n k1 k with # x p0, 0q y p0, 0q 0, n ¥ 1 , n n (see [2, 3]). (112) x0pt, sq y0pt, sq 1, @t, s ,

7 Equations pT q are the 2–Toda equations of Ueno–Takasaki [55], but equations pT q with precisely the initial conditions (106) and (107) are an invariant subsystem of the 2-Toda equations which are equivalent to the Toeplitz lattice, (110), (111). The latter equations are Hamiltonian, with ω the symplectic form. 30 Mark Adler

Example

Bxn Byn p1 x y qx , p1 x y qy , Bt n n n 1 Bt n n n 1 1 1 (113) Bxn Byn p1 xnynqxn1 , p1 xnynqyn1 , Bs1 Bs1 yielding the

Ladik–Ablowitz lattice: B B B . (114) Bt Bt1 Bs1

2.4 Virasoro constraints and diﬀerence relations

In this section, using a 2–Toda vertex operator, which generates a subspace of the tangent space of the sequence of 2–Toda τ-functions tτnpt, squ, we derive Virasoro relations in our special case. Indeed, deriving a Lax-equation for the vertex operator leads to a fixed point theorem for our particular sequence of matrix integral τ-functions, fixed under an integrated version of the vertex operator, integrated over the unit circle. The Virasoro relations coupled with the Toeplitz lattice identities then lead to difference relations after some manipulation. We present the following operators:

Toda vertex operator:

p q p q X u, v fn t, s n¥0 ° 8 i i ptiu siv q n1 1 1 : pe 1 puvq fn1pt ru s, s rv sqn¥0 . (115)

Integrated version: » du Y γ : uγ Xpu, u1q . (116) S1 2πiu Virasoro operator:

γ p γ q p2qp q p2qp q p q p1qp q p q p1qp q Vk Vk,n n¥0 : Jk t Jk s k γ θJk t 1 θ Jk s (vector diﬀerential operator in t, s, acting diagonally as deﬁned explicitly in (123), (124), and (125)) . (117) Integrable Systems, Random Matrices and Random Processes 31

Main facts:

d Lax-equation: uγ u ukγ Xpu, u1qq rVγ ,Xpu, u1qs , (118) du k Commutativity: rVγ ,Y γ s 0 , (119) k γ pγqp q Fixed Point: Y I I,I : n!τn t, s n¥0 , (120) where » ¹n ° p q 1 γ 8 p j j q dzk γ p q | p q|2 j1 tj zk sj zk τn t, s ∆n z zk e , n! p 1qn 2πiz S k1 k

pτ0 1q (121)

pγq and tτn pt, squ satisfy an s`p2, Zq algebra of Virasoro constraints: $ &'k 1, θ 0 , p q Vγ τ γ pt, sq 0 for k 0, θ arbitrary , pn ¥ 0q . (122) k,n n %' k 1, θ 1,

Proof of main facts.

The proof of the Lax-equation is a lengthy calculation (see [3]). Integrating the Lax-equation with regard to » du uγ

S1 2πiu immediately leads to the commutativity statement. To see the ﬁxed point statement, compute (setting u zn) I pt, sq n!τ pγqpt, sq n » n γ ° u du 8pt uj s uj q n1 n1 e 1 j j u u S1 2πiu » ¹n1 ° γ z u 8 j j z dz k ptj z sj z q k k ∆n1pzq∆n1pzq 1 1 e 1 k k p 1qn1 u zk 2πizk » S k1 γ ° ° u du 8pt uj s uj q 1 n1 8puj {j B{Bt uj {j B{Bs q e 1 j j puu q e 1 j j S1 2πiu » n¹1 ° γ 8 j j z dz ptj z sj z q k k ∆n1pzq∆n1pzq e 1 k k p 1qn1 2πiz S k1 k γ pY Iqn , yielding the ﬁxed point statement. Finally, to see the s`p2,Zq Virasoro statement, observe that from the other main facts, we have 32 Mark Adler

pr γ p γ qns q 0 Vk, Y I n p γ p γ qn p γ qn γ q Vk Y I Y VkI n pVγ I pY γ qnVγ Iq k » k n ° ° γ du 8p j j q 8p j { B{B j { B{B q γ 1 tj u sj u 1 u j tj u j sj Vk,nIn u e e » S1 2πiu ° ° du 8p j j q 8p j { B{B j { B{B q γ γ 1 tj u sj u 1 u j tj u j sj u e e Vk I0 , S1 2πiu

γ upon using the backward shift (see (115)) present in Y . Note I0 1, and γ so it follows from the explicit Virasoro formulas given below, that Vk,01 0 precisely if k 1, 0, 1 and θ is as speciﬁed in (122), yielding the Virasoro constraints (122). We now make explicit the Virasoro relations.

Explicit Virasoro formulas:

$ & p1q p p1q q Jk Jk nδ0k n¥0 , (123) % p2q 1 p p2q p q p1q p q q Jk 2 Jk 2n k 1 Jk n n 1 δ0k n¥0 . $ B ' p1q &J pkqt k B k tk ' ° B , if k 0, 1 . (124) %' p2q Jk 2 iti Btik Virasoro relations: $ B B ' γ pγq 1 p2qp q 1 p2qp q pγq 'V1,nτn J1 t J1 s n t1 γ τn 0 , ' 2 2 Bs1 Bs1 &' γ pγq 1 p2q 1 p2q pγq V τn J ptq J psqnγ τ 0 , (125) ' 0,n 2 0 2 0 n ' ' B B %' γ pγq 1 p2qp q 1 p2qp q pγq V1,nτn J1 s J1 t n s1 γ τn 0 . 2 2 Bt1 Bt1 We now derive the ﬁrst diﬀerence relation (the second is similar) in Case 2 of Theorem 2.1. Setting for arbitrary a, b, c, t, and s,

αiptq : api 1qti1 biti cpi 1qti1 cpn γqδi1 , (126) β psq : api 1qs bis cpi 1qs apn γqδ , i ¸ i 1 i i 1 ¸ i1 pnq p q i pnq p q i L1 : αi t L1 and L2 : βi t L2 , (127) i¥1 i¥1 and remembering p q p qn τn τn xn, yn 1 0 , 0 , vn 1 xnyn , τn τn Integrable Systems, Random Matrices and Random Processes 33 compute, using the Virasoro relations and the Toeplitz ﬂow: " xnyn 1 p q 1 p q 0 aV1,n bV0,n cV1,n τn aV1,n bV0,n cV1,n τn vn τn τn * 2 paV1,n bV0,n cV1,nqτn τn ¸ xnyn B B aiptq βipsq log xnyn vn Bti Bsi i¥1 B B x c a log n Bt1 Bs1 yn ¸ B B xnyn p q ai B βi B log xn log yn vn ¥ ti si i 1 B B c a plog xn log ynq Bt1 Bs1 p pnq pnq q L1 L2 aL2 cL1 nn p pnq pnq q L1 L2 aL2 cL1 n1,n1 , B p pnq pnqq p q n L1 L2 n,n aL2 cL1 nn . (128) We then ﬁnd our result by specializing the above identity to the precise locus L in t, s space corresponding to the measure ρpzq of Case 2 of Theorem 2.1: $ # , 1 i 1 i ' p q u pγ d γ d q, for 1 ¤ i ¤ N / 'it it 0 : i 1 1 2 2 1 / & i i p 1 i 1 i q ¤ 8 . γ1d1 γ2d2 , for N1 1 i L : # (129) ' 2 i 2 i / ' p q u pγ d γ d q, for 1 ¤ i ¤ N / %'is is 0 : i 1 1 2 2 2 -/ i i p 2 i 2 iq ¤ 8 γ1 d1 γ2 d2 , for N2 1 i with 1 pa, b, cq ? p1, d1 d2, d1d2q , (130) d1d2 and so in (129) all αiptq 0 for all i ¥ N1 2 and all βiptq 0 for i ¥ N2 2.

2.5 Singularity conﬁnement of recursion relations

p0q Since for the combinatorial examples the unitary integral In ptq satisfies Painlevédifferential equations in t, it is natural to expect they satisfy a discrete version of the Painlevéproperty regarding the development of poles. For instance, algebraically integrable systems (a.c.i.) z9 fpzq, z P Cn, admit Laurent solutions depending on the maximal number, n1, of free parameters [see [14]). An analogous property for rational recursion relations

k zn F pzn1, . . . , znδq, zn P C , 34 Mark Adler would be that there exists solutions of the recursion relation tzipλqu which are “formal Laurent” solutions in λ developing a pole which disappears after a ﬁnite number of steps, and such that these “formal Laurent” solutions depend on the maximal number of free parameters δ k (counting λ) and moreover the coeﬃcients of the expansions depend rationally on these free parameters. We shall give results for Case 2 of Theorem 2.1 and 2.2, where N1 N2 N. The results in this section, Theorem 2.3–2.6, are due to Adler–van Moerbeke– Vanhaecke and can be found with proofs in [13].

Self-dual case: ° N u pziziq{i ρpzq e 1 i . (131)

Theorem 2.3 (Singularity conﬁnement: self-dual case). For any n P 8 Z, the diﬀerence equations Γkpxq 0, pk P Zq admit two formal Laurent p q solution x xk λ kPZ in a parameter λ, having a (simple) pole at k n only and λ 0. These solutions depend on 2N non-zero free parameters

α pan2N , . . . , an2q and λ .

Explicitly, these series with coeﬃcients rational in α are given by pε 1q: ¸8 p q piqp q i xk λ xk α λ , k n 2N, i0

xkpλq αk , n 2N ¤ k ¤ n 2 ,

xn1pλq ε λ , 8 1 ¸ x pλq xpiqpαqλi , n λ n i0 ¸8 p q piq p q i xn1 λ ε xn1 α λ , i1 ¸8 p q piqp q i xk λ xk α λ , n 1 k . i0 General case: ° N i i pu z u z q{i ρpzq e 1 i i .

Theorem 2.4 (Singularity conﬁnement: general case). For any n P Z, ˜ the diﬀerence equations Γkpx, yq Γkpx, yq 0, pk P Zq admit a formal p q p q Laurent solution x xk λ kPZ and y yk λ kPZ in a parameter λ, having

8 We consider the obvious bi-infinite extension of Lipx, yq (89) which through (98) r defines a bi-infinite extension of Γkpx, yq, Γkpx, yq. Integrable Systems, Random Matrices and Random Processes 35

a (simple) pole at k n and λ 0, and no other singularities. These solutions depend on 4N non-zero free parameters

αn2N , . . . , αn2, αn1, βn2N , . . . , βn2 and λ .

Setting zn : pxn, ynq and γi : pαi, βiq, and γ : pγn2N , . . . , γn2, αn1q, the explicit series with coeﬃcients rational in γ read as follows: ¸8 p q piqp q i zk λ zk γ λ , k n 2N, i0

zkpλq γk , n 2N ¤ k ¤ n 2 ,

xn1pλq αn1 , 1 yn1pλq λ , αn1 8 1 ¸ z pλq zpiqpγqλi , n λ n i0 ¸8 p q piqp q i zk λ, γ zk γ λ , n k . i0 Singularity confinement is consequence of : (1) Painlevéproperty of a.c.i. Toeplitz lattice. (2) Rational difference relations as a whole define an invariant manifold of the Toeplitz lattice. (3) Formal Laurent solutions of Toeplitz lattice with maximal parameters restrict to the above invariant manifold, restricting the parameters. (4) Reparametrizing the “restricted” Laurent solutions by t Ñ λ and “restricted parameters” Ñ γ yields the confinement result. We discuss (1) and (2). Indeed, consider the Toeplitz lattice with the p1q p2q Hamiltonian H H1 H1 , yielding the flow of (114): General case: B xk p qp q B 1 xkyk xk1 xk1 , t k P Z . Byk p1 x y qpy x q , Bt k k k 1 k 1 Self-dual case: B xk 2 p1 x qpx x q , k P Z . Bt k k 1 k 1 Then we have 36 Mark Adler

Maximal formal Laurent solutions: Theorem 2.5. For arbitrary but fixed n, the first Toeplitz lattice vector field (114) admits the following formal Laurent solutions, p q 1 p q p 2q xn t p q an1an1 1 at O t , an1 an1 t 1 an1a an1a 2 ynptq 1 a t Opt q , pan1 an1qt an1 an1 2 xn1ptq an1 an1at Opt q ,

1 at 2 yn1ptq Opt q an1 an 1 whereas for all remaining k such that |k n| ¥ 2,

2 xkptq ak p1 akbkqpak1 ak1qt Opt q , 2 ykptq bk p1 akbkqpbk1 bk1qt Opt q , where a, a, an1 and all ai, bi, with i P Zztn 1, n, n 1u and with bn1 1{an1, are arbitrary free parameters, and with pan1 an1qan1an1 0. In the self-dual case it admits the following two formal Laurent solutions, parametrized by ε 1, ε 2 xnptq 1 pa aqt Opt q , 2t 2 x ptq ε 1 4at Opt q , n 1 p q p 2 qp q p 2q | | ¥ xk t ε ak 1 ak ak1 ak1 t O t , k n 2 , where a, a and all ai, with i P Zztn1, n, n1u are arbitrary free parameter and an1 an1 1. Together with time t these parameters are in bijection with the phase space variables; we can put for the general Toeplitz lattice for example zk Ø pak, bkq for |k n| ¥ 1 and xn1 Ø an1 and yn1, xn, yn Ø a, a, t. Consider the locus Lˆ deﬁned by the diﬀerence relations (98) of Case 3 of Theorem 2.1, namely: General case: r Lˆ tpx, yq | Γnpx, y, uq 0 , Γnpx, y, uq 0 , @nu . Self-dual case:

Lˆ tx | Γnpx, y, uq 0, @nu , r where we now explicitly exhibit the dependence of Γn, Γn on the coeﬃcients of the measure, namely u. The point of the following theorem is that Lˆ is p1q p2q an invariant manifold for the ﬂow generated by H H1 H1 , upon our imposing the following u dependence on t pvn 1 xnynq. Integrable Systems, Random Matrices and Random Processes 37

Theorem 2.6. Upon setting Bui{Bt δ1i, the recursion relations satisfy the following diﬀerential equations 9 r Γk vkpΓk1 Γk1q pxk1 xk1qpxkΓk ykΓkq , r9 r r r Γk vkpΓk1 Γk1q pyk1 yk1qpxkΓk ykΓkq , which specialize in the self-dual case to 9 Γk vkpΓk1 Γk1q . Sketch of proof :

The proof is based on the crucial identities: r Γn V0xn nxn and Γn V0yn nyn , where ¸ B B V : u u , 0 i Bt i Bs i¥1 i i and B B B "xn* "yn* "uk* vnxn1 , vny 1 , δk,1 , B t1 B t1 B t1 s1 s1 s1 which implies B "Γn* r vnΓn1 xn1pxnΓn ynΓnq , B t1 s1 B r "Γn* r r vnΓn 1 yn 1pxnΓn ynΓnq , B t1 s1 which, upon using B{Bt B{Bt1 B{Bs1, yields the theorem.

3 Coupled Random Matrices and the 2–Toda Lattice

3.1 Main results for coupled random matrices

The study of coupled random matrices will lead us to the 2–Toda lattice and bi-orthogonal polynomials, which are essentially 2 of the 4 wave functions for the 2–Toda lattice. This problem will lead to many techniques which will come up again, as well as a PDE for the basic probability in coupled random matrices. Let M1,M2 P Hn, Hermitian n n matrices and consider the probability ensemble of 38 Mark Adler

Coupled random matrices:

³ 2 2 1{2 TrpM M 2cM1M2q dM1 dM2e 1 2 P pM ,M q S ³ S (132) 1 2 1{2 TrpM 2M 22cM M q dM1 dM2e 1 2 1 2 HnHn with ¹n ¹n 2 p q 2 p q dM1 ∆n x dxi dU1, dM2 ∆n y dyi dU2 . 1 1 rr s sr s Given E E1 E2 1 a2i1, a2i 1 b2i1, b2i , deﬁne the boundary operators: 1 ¸2r B ¸2s B ¸2r B B A c , A a c , 1 1 c2 Ba Bb 2 j Ba Bc 1 j 1 j 1 j (133) B1 A1 , B2 A2 , aØb aØb which form a Lie algebra:

1 c2 2c rA , B s 0 , rA , A s A , rA , B s A , 1 1 1 2 1 c2 1 2 1 1 c2 1 (134) 1 c2 2c rA , B s 0 , rB , B s B , rB , A s B , 2 2 1 2 1 c2 1 2 1 1 c2 1 We can now state the main theorem of Section 3:

Theorem 3.1 (Adler–van Moerbeke [4]). The statistics

1 1 F : log P pEq : log P (all (M –eigenvalues) P E , n n n n 1 1 all (M2–eigenvalues) P E2) (135) satisﬁes the third order nonlinear PDE: B2A1Fn A2B1Fn A1 2 B1 2 0 . (136) B1A1Fn c{p1 c q A1B1Fn c{p1 c q

In particular for E p8, as p8, bs, setting: x : a cb, y : ac b, A1 Ñ B{Bx, B1 Ñ B{By, (136) becomes 2 2 2 2 B pc 1q B Fn{BxBc 2cx p1 c qy Bx pc2 1qB2F {BxBy c n B p 2 q2B2 {B B p 2q c 1 Fn y c 2cy 1 c x 2 2 . (137) By pc 1qB Fn{ByBx c Integrable Systems, Random Matrices and Random Processes 39

3.2 Link with the 2–Toda hierarchy

In this section we deform the coupled random matrix problem in a natural way to introduce the 2–Toda hierarchy into the problem. We ﬁrst need the celebrated Harish–Chandra–Itzkson–Zuber formula [23]: HCIZ: x diagpx1, . . . , xnq, y diagpy1, . . . , ynq » npn1q{2 p cxiyj q c Tr xUyU 1 2π det e 1¤i,j¤n dU e p q p q . (138) Upnq c n!∆n x ∆n y Compute, using HCIZ: ¼ ° 8 i i c Tr M1M2 Trp ptiM siM qq dM1 dM2e e 1 1 2 (t, s deformation)

HnpE1qHnpE2q » # + ¹n ° 8p i i q 2 p q 2 p q 1 tixk siyk ∆n x ∆n y e dxk dyk EnEn 1 2 ¼ k1 c Tr U U 1U yU 1 dU dU e 1 1 2 2 1 2

UpnqUpnq » # + ¹n ° 8p i i q 2 p q 2 p q 1 tixk siyk ∆n x ∆n y e dxk dyk EnEn 1 2 » »k1 1 1 c Tr U U2yU U1 dU1 e 1 2 dU2 Upnq Upnq » » # + ¹n ° 8p i i q 2 p q 2 p q 1 tixk siyk ∆n x ∆n y e dxk dyk En En 1 2 » » k1 1 1 c Tr UyU U U1 U2 dU1 e dU Upnq Upnq dU dU2 » » ¹n ° 8 i i cxiyj ptix siy q cn ∆npxq∆npyqpdet e q1¤i, e 1 k k dxk dyk n n j¤n E1 E2 1 » » ¸ ¹n ¹n σ cxiyσpiq : cn ∆npxq∆npyq p1q e dµpxkq dψpykq En En 1 2 σPSn 1 1 n n (setting: yσpiq Ñ yi, E Ñ E ) » » 2 2 ¹n ¹n cxiyi n!cn ∆npxq∆npyq e dµpxkqdψpykq En En » 1 2 1 1 ¹n ° 8 i i ptix siy qcxkyk n!cn ∆npxq∆npyq e 1 k k dxk dyk,E E1 E2 , n E k1 where we have used in the above that Haar measure is translation invariant. We now make a further c-deformation of this matrix integral. 40 Mark Adler

Deﬁne τ-function:

τnpt, s, C, Eq » ¹n ° ° 1 8 i i i j ptix siy q ¥ cij x y : ∆npxq∆npyq e 1 k k i,j 1 k k dxk dyk , n! n E k1

pτ0 1q which is not a matrix integral ! (139) Thus τnpt, s, C, Eq is t, s, C deformation of PnpEq . τnpt, s, C, Rq It is quite crucial to recast the τ-function using the de Bruijn trick: Moment matrix form of τ-function:

τnpt, s, c, Eq » ¹n ° ° 1 8 i i i i ptix siy q ¥ cij x y ∆npxq∆npyq e 1 k k i,j 1 k k dxk dyk n! En » k 1 1 ¹n det fipxjq 1¤i, det gipyjq 1¤i, dψpxk, ykq n! n ¤ ¤ E j n j n 1 i1 pfipxq gipxq x q » ¸ ¹n 1 σµ p1q fσpiqpxiqgipyµpiqqdψpxµpiq, yµpiqq n! n σ,µPS E 1 n » ¸ ¹n 1 σµ p1q fσµpiqpxµpiqqgipyµpiqqdψpxµpiq, yµpiqq n! n σ,µPS E 1 n » ¸ ¸ ¹n 1 σ1 p1q fσ1piqpxqgipyqdψpx, yq n! P 1P E µ Sn σ Sn » 1 1 ¸ det fipxqgjpyqdψpx, yq 1¤i, n! P E µ Sn j¤n

detpµij q0¤i, , (140) j¤n1 with » ° ° 8 k k α β i j ptkx sky q ¥ cαβ x y i j µijpt, s, C, Eq x y e k 1 k α,β 1 dx dy :xx , y y . (141) E Thus we have shown:

τnpt, s, C, Eq det mn , mn pµij q0¤i, . (142) j¤n1 This immediately leads to the Integrable Systems, Random Matrices and Random Processes 41

2–Toda diﬀerential equations – Moment form:

Bµij Bµij µik,j , µi,jk , (143) Btk Bsk which we reformulate in terms of the moment matrix m8 B B m8 k m8 T k Λ m8 , m8pΛ q , (144) Btk Bsk or equivalently ° ° 8 t Λk 8 s pΛT qk m8pt, sq e 1 k m8p0, 0qe 1 k , (145) with the semi-inﬁnite shift matrix 0 1 0 0 ... 0 0 1 0 ... 0 0 0 1 ... Λ : . (146) 0 0 0 0 ......

Thus p q p q p p q p q T p q τn t, s, C, E det mn t, s det En t m8 0, 0 En s , (147) with 2 Enptq pI s1ptqΛ s2ptqΛ qn8 and ° ¸8 8 i tiz i e 1 : siptqz , (148) 0 with siptq the elementary Schur polynomials.

3.3 L U decomposition of the moment matrix, bi-orthogonal polynomials and 2–Toda wave operators

The L U decomposition of m8 is equivalent to bi-orthogonal polynomials; indeed, consider the L U decomposition of m8, as follows (see [9]): 1p q p q 1p T q T 1 m8 LhU : S m8 h m8 S m8 : S1 S2 (149) where we deﬁne the string orthogonal polynomials 42 Mark Adler 1 m y det n . . 1 . y n p1q µn0 . . . µn,n1 y p pyq : Spm8q 2 : (150) n n¥0 y det m . n . .

n¥0 and 1 p q y p 2 pyq : SpmT q 2 . (151) n n¥0 8 y . . Setting i j x , y: xx , y y : µij pt, s, Cq , (152) conclude (as a tautology) the deﬁning relations of the monic bi-orthogonal polynomials, namely x p1q p2qy ðñ p q p T q T p q pi , pj hiδij S m8 m8 S m8 h m8 , (153) with the ﬁrst identity a consequence of (150) and (151), which also implies det mi1 hpm8q : diag h0, h1, . . . , hi ,... , (154) det mi and by (149) 1 T T S1 Spm8q ,S2 hpm8q S pm8q . (155)

We now deﬁne the 2–Toda operators:

1 T 1 L1 S1ΛS1 ,L2 S2Λ S2 . (156) 1 Since m8 S1 S2, with

S1 P I g ,S2 P g , then 9 1 P 9 1 P S1S1 g , S2S2 g , with g strictly lower triangular matrices and g upper triangular matrices, including the diagonal, and g g g : all semi-inﬁnite matrices. Compute

9 1 p 1 q9 1 9 1 9 1 P S1m8S2 S1 S1 S2 S2 S1S1 S2S2 g g , (157) Integrable Systems, Random Matrices and Random Processes 43

p 1q9 1 9 1 where we have used S1 S1 S1S1 . On the other hand, one computes, using (144), for B{Btn or B{Bsn separately, that: B m8 1 n 1 n 1 1 n 1 S1 S2 S1Λ m8S2 S1Λ S1 S2S2 S1Λ S1 Btn n p nq p nq P L1 : L1 L1 g g , (158) and B m8 1 p T qn 1 1 p T qn 1 S1 S2 S1m8 Λ S2 S1S1 S2 Λ S2 Bsn p T qn 1 S2 Λ S2 n p nq p nq P L2 : L2 L2 g g , (159) and so (157), (158), and (159) yield the diﬀerential equations B B S1 1 p nq S2 1 p nq S1 L1 , S2 L1 , Btn Btn B B (160) S1 1 p nq S2 1 p nq S1 L2 , S2 L2 . Bsn Bsn

Setting χpxq p1, z, z2,... qT we now connect the bi-orthogonal polynomials with the 2–Toda wave functions:

2–Toda wave functions: $ ° ° & 8 t zk p1q 8 t zk Ψ1pzq : e 1 k p pzq e 1 k S1χpzq , ° ° % 8 k p q 8 k 1 p q 1 skz 1 2 p 1q 1 skz p qT p 1q Ψ2 z : e h p z e S2 χ z .

Eigenfunction identities:

p q p q T p q 1 p q L1Ψ1 z zΨ1 z ,L2 Ψ2 z z Ψ2 z . Formulas (160) and (156) yield, t s ﬂows for Li and Ψ: $ B B ' Li rp nq s Li rp nq s ' L1 ,Li , L2 ,Li , i 1, 2, n 1, 2,... ' Btn Bsn ' & B B Ψ1 p nq Ψ1 p nq ' B L1 Ψ1 , B L2 Ψ1 , ' tn sn ' ' B B %' Ψ2 p nqT Ψ2 p nqT L1 Ψ1 , L2 Ψ2 . Btn Bsn 44 Mark Adler

Wave operators: ° ° 8 k 8 T k tkΛ skpΛ q W1 : S1e 1 ,W2 : S2e 1 , (161) satisfy 1 1 1 1 1 1 1 1 W1pt, sqW1pt , s q W2pt, sqW2pt , s q , @t, s, t , s . (162) All the data in 2–Toda is parametrized by τ-functions, to wit: L1, L2, Ψ1, Ψ2 parametrized by τ-functions: $ p q ' Ψ1 z, t, s ' ' p r 1s q °8 ' τn t z , s t zi n &' e 1 i z τnpt, sq n¥0 , rxs px, x2{2,... q , (163) ' p q ' Ψ2 z, t, s ' ° ' p r sq 8 ' τn t, s z s z i n % e 1 i z τ pt, sq $ n 1 n¥0 ¸8 r ' s pB qτ τ ' k ` t n k ` 1 n k` 'L1 diag Λ , & τnk`1τn `0 n¥0 (164) ' r ' °8 s pB qτ τ 'p p T qk 1q ` t n k ` 1 n % h L2 h `0 diag , τnk`1τn r r BtÑBs with r B 1 B 1 B s`ptq the elementary Schur polynomials, Bt , , ,... , Bt1 2 Bt2 3 Bt3 and the Hirota symbol: B B p q p q p B f g : p B f t y g t y . (165) t y y0

3.4 Bilinear identities and τ -function PDE’s Just like in KP theory (see Section 1.3), where the bilinear identity generates the KP hierarchy of PDE’s for the τ-function, the same situation holds for the 2–Toda Lattice. In general 2–Toda theory (see [17]) the bilinear identity is a consequence of (163) and (162), but in the special case of 2–Toda being generated from bi-orthogonal polynomials, we can and will, at the end of this section, sketch a quick direct alternate proof of Adler–van Moerbek– Vanhaecke [15] based on the bi-orthogonal polynomials, which has been vastly generalized. Since all we ever need of integrability in any problem is the PDE hierarchy, it is clearly of great practical use to have in general a quick proof of just the bilinear identities, but without all the usual integrable baggage. We now give the Integrable Systems, Random Matrices and Random Processes 45

2–Toda bilinear identities: ¾ ° 8 1 i 1 1 1 1 ptit qz nm1 τnpt rz s, sqτm1pt rz s, s qe 1 i z dz z8 ¾ ° 8 1 i 1 1 1 1 psis qz mn1 τn1pt, s rz sqτmpt , s rz sqe 1 i z dz , z0 @s, t, s1, t1, m, n . (166)

The identities are a consequence of (162) and (163) and they yield, as in Section 1.3 (see Appendix) a generating function involving elementary Schur 9 polynomials sjpq and arbitrary parameters a, b, in the following Hirota form:

8 ¸ °8 pakB{BtkbkB{Bskq 0 smnjp2aqsjpB˜tqe 1 τm1 τn j0 8 ¸ °8 pakB{BtkbkB{Bskq smnjp2bqsjpB˜sqe 1 τm τn1 (167) j 0 B2 pB˜ q p 2 q aj1 2sj t τn2 τn τn1 τn1 0 aj1 , (168) Bs1Btj1 upon setting m n 1, and all bk, ak 0, except aj1. Note: Bf Bg s0ptq 1 , s1ptq t1 , s1pB˜tqf g g f , Bt1 Bt1 and skptq tk poly. pt1, . . . , tk1q. This immediately yields the 2–Toda τ-function identities:

2 B sk1pB˜tqτn2 τn log τ (169) B B n 1 2 s1 tk τn1 $ τ τ ' n 2 n & 2 , k 1 , τn1 (170) 'τ τ B τ %' n 2 n log n 2 , k 2 , 2 B τn1 t1 τn from which we deduce, by forming the ratio of the k 1, 2 identities, and using (164):

9 Hopefully there will be no confusion in this section between the elementary Schur polynomials, sj pq, which are functions, and the time variables sj , which are parameters, but the situation is not ideal. 46 Mark Adler

Fundamental identities: B B2{B B 2 τn1 s1 t2 log τn p q L1 n 1,n log 2 , (171) Bt1 τn1 B {Bs1Bt1 log τn Ø Ø T 1 (and by duality t s, L1 hL2 h ) B B2{B B p T 1q2 τn1 t1 s2 log τn hL2 h n1,n log 2 . (172) Bs1 τn1 B {Bt1Bs1 log τn As promised we now give, following Adler–van Moerbeke–Vanhaecke [15]:

Sketch of alternate proof of bilinear identities:

The proof is based on the following identities concerning the bi-orthogonal polynomials and their Cauchy transforms with regard to the measure dρ deﬁn- ing the moments of (141) and (152) µij : ° ° 8pt xis yiq c xαyβ dρpx, y, t, s, cq e 1 i i α,β¥1 αβ dx dy .

Namely, the bi-orthogonal polynomials of (150) and (151) and their formal Cauchy transforms with regard to dρ can be expressed in terms of τ-functions as follows:10 $ p r 1s q ' p1q n τn t z , s 'pn pz, t, sq z , ' τ pt, sq ' n ' piq ' (suppressing c and E in pn and τn) , ' ' 1 & p q τ pt, s rz sq p 2 pz, t, sq zm m , m τ pt, sq (173) 'B F m ' 1 ' p q 1 τ pt, s rz sq ' p 1 pxq, zn1 n 1 , ' n z y τ pt, sq 'B F n ' 1 ' 1 τ pt rz s, sq %' p2qp q m1 m 1 , pm y z . z x τmpt, sq These formulas are not hard to prove, they depend on substituting ° 1 ¸ i i x x x e i¥1 px{zq {i 1 or 1 (174) z z z i¥0 into formula (142), which express τnpt, s, C, Eq in terms of the moments µijpt, s, c, Eq of (141). Then one must expand the rows or columns of the ensuing determinants using (174) and make the identiﬁcation of (173), using (150) and (151), an amusing exercise. If one then substitutes (173) into the bilinear identity (166) divided by τnpt, sqτmpt, sq, thus eliminating τ-functions, and if we make use of the following self-evident formal residue identities: ° 10 {p q { 8 p { qi {p q 1 z x : 1 z i0 x z , etc., 1 z y , and thus z is viewed as large. Integrable Systems, Random Matrices and Random Processes 47 ¾ B F hpxq dz fpzq , gpyq xfpxqhpxq, gpyqy , z x 2πi 8 z ¾ B F (175) hpyq dz fpzq gpxq, xgpxq, fpyqhpyqy , z y 2πi z8 with ¸8 i fpzq aiz , (176) i0 the bilinear identity becomes a tautology.

3.5 Virasoro constraints for the τ -functions

In this section we derive the Virasoro constraints for our τ-functions τpt, s, C, Eq using their integral form:

p1q p q p2q p q ¥ Vk τ t, s, C, E 0 ,Vk τ t, s, C, E 0 , k 1 . (177) Explicitly:

$ '¸r B ¸ B ' k1 E p2q p q E E &' ai τn Jk,n t icij τn : Vkτn Ba Bc 1 i i,j¥1 i k,j '¸s B ¸ B , ' k1 E p2q p q E r E %' bi τn Jk,n s jcij τn : Vkτn Bb Bc 1 i i,j¥1 i,j k k ¥ 1 , n ¥ 0 , (178) with ¤r ¤s E ra2i1, a2is rb2i1, b2is , (179) 1 1 E τ τnpt, s, C, Eq n » ¹n ° 1 8 i tix ∆npxq e 1 k dxk n! En k 1 ¹n ° ¹n ° 8 i i j siy ¥ cij x y ∆npyq e 1 k dyk e i,j 1 k k , (180) k1 k1 and 48 Mark Adler p1qp q p1q p q p p1qp q p0qq Jk t : Jk,n t ¥ : Jk t nJk n¥0 , n 0 p2qp q p2q p q Jk t : Jk,n t n¥0 1p p2qp q p q p1qp q p q p0qq : Jk t 2n k 1 Jk t n n 1 Jk n¥0 , 2 (181) B p1qp q p q p0q Jk t : k tk,Jk δ0k , Btk ¸ B2 ¸ B ¸ p2qp q p qp q Jk t : 2 iti iti jtj . Bt Bt Bt ijk i j i¥1 i k ijk

Main fact:

p2q Jk forms a Virasoro algebra of charge c 2,

3 p2q p2q p2q pk kq rJ , J s pk `qJ p2q δ . (182) k ` k` 12 k, ` To prove (178) we need the following lemma:

Lemma 3.1 (Adler–van Moerbeke [5]). Given ° 1 8 i V ρ 1 g 0 βiz ρ e with V °8 , i ρ f 0 αiz the integrand ¹n ° 8 i tix dInpxq : ∆npxq pe 1 ρpxkqdxkq , (183) k1 satisﬁes the following variational formula: ¸8 d p2q p1q dI px ÞÑ x εfpx qxm 1q pα J β J q dI . (184) dε n i i i i ` m`,n ` m`1,n n ε0 `0 ± n The contribution coming from 1 dxj is given by ¸8 p q p1q a` ` m 1 Jm`,n dIn . (185) `0

Proof of (178).

Ñ k1 ¤ ¤ First make the change of coordinates xi xi εxi , 1 i n, in the integral (180), which remains unchanged, and then diﬀerentiate the results by ε, at ε 0, which of course yields 0, i.e., d E τn Ñ k1 0 . (186) xi xi εxi dε ε0 Integrable Systems, Random Matrices and Random Processes 49

Now, by (180), there are precisely 3 contributions to the `.h.s. of (186), namely p q p2q p q E one of the form (184), with ρ x 1, yielding Jk,n t τn , one coming from ¹n ° d c xi yj e i,j¥1 ij ` ` dε k1 xsÑxsεxs ` 1 ε0 ¸ ¸n ¹n ° j i j i k i,j¥1 cij x`y` icij x` y` e i,j¥1 `1 `1 ¸ ¹n ° B i j ¥ cij x y icij e i,j 1 ` ` , Bc i,j¥1 i k,j `1 ° B{B E 11 yielding i,j¥1 icij cik,j τn . Finally, we have a third contribution, since the limits of integration the integral (180) must change: Ñ k1 p 2q ¤ ¤ a i ai εai 0 ε , 1 i 2r .

E Upon diﬀerentiating τn with respect to the ε in these°new limits of integra- 2r k1B{B E tion, we have by the chain rule, the contribution 1 ai aiτn . Thus altogether we have:

¸2r B ¸ B d E k1 E p2q E E p q 0 τn x Ñx εxk 1 ai τn Jk,n t τn icij τn , dε i i i Ba Bc ε0 1 i i,j¥1 i k,n yielding the ﬁrst expression (178). The second expression follows from the ﬁrst by duality, t Ø s, a Ø b, cij Ø cji.

3.6 Consequences of the Virasoro relations Observe from (132), (139) that e2 p0, 1, 0, 0,... q τ Ept e {2, s e {2,Cq P pEq n 2 2 , (187) n Rp { { q τn t e2 2, s e2 2,C L L : tt s 0, all cij 0, but c11 cu , (188)

Ep 1 1 q and so computing (178) for τn t 2 e2, s 2 e2,C requires us to shift the t, s p2q p q p2q p q in Jk,n t , Jk,n s accordingly, and we ﬁnd from (181) shifted, the following:

Akτn Vkτn, Bkτn Wkτn, k 1, 2 (189) with E 1 1 τn τn t 2 e2, s 2 e2,C , and 11 It must be noted that: B{pBc0,nq B{pBsnq, B{pBcn,0q B{pBtnq. 50 Mark Adler ¸2r B ¸2s B ¸2s B ¸2r B 1 1 A1 2 c , B1 2 c , 1c Baj Bbj 1c Bbj Baj 1 1 1 1 (190) ¸2r B ¸2s B A a , B b , 2 j Ba 2 j Bb 1 j 1 j with 1 r p V : pV cV qV v 1 1 c2 1 1 1 1 B npt cs q : 1 1 B 2 t1 c 1 ¸ ¸ 1 B B B B i ti csi cij i jc , c2 1 Bt Bs Bc Bc i¥2 i 1 i 1 i,j¥1, i 1,j i,j 1 i,jp1,1q 1 r x W : pcV V qW w 1 1 c2 1 1 1 1 B npct s q : 1 1 B 2 s1 c 1 ¸ ¸ 1 B B B B i cti si cij ci j , c2 1 Bt Bs Bc Bc i¥2 i 1 i 1 i,j¥1, i 1,j i,j 1 i,jp1,1q B V : V c :Vp v 2 0 Bc 2 2 B ¸ B npn 1q ¸ B : it ic , Bt i Bt 2 ij Bc 2 i¥1 i i,j¥1, ij pi,jqp1,1q B W :Vr c :Wx w 2 0 Bc 2 2 B ¸ B npn 1q ¸ B : is jc . Bs i Bs 2 ij Bc 2 i¥1 i i,j¥1, ij pi,jqp1,1q p x p x Note V1, W1, V2, W2 are ﬁrst order operators such that (and this is the point): B B B B p x p x V1 , W1 , V2 , W2 , (191) Bt1 Bs1 Bt2 Bs2 L L L L and npt cs q npct s q npn 1q v 1 1 , w 1 1 , v w . (192) 1 1 c2 1 1 c2 2 2 2 Because of (191) we call this a principal symbol analysis. Hence Integrable Systems, Random Matrices and Random Processes 51 p x Ak log τn Vk log τnvk , Bk log τn Wk log τnwk , k 1, 2 , (193) and so on the locus L using (191) – (193) we ﬁnd: B B B log τn A1 log τn , B log τn B1 log τn , t1 L L s1 L L B B (194) B log τn A2 log τn , B log τn B2 log τn t2 L L s2 L L npn 1q npn 1q . 2 2 Extending this analysis to second derivatives, compute: p p B1A1 log τn B1pV1 log τn v1q B1V1 log τn B1pv1q L L L L pxq p V1B1 log τn B1pv1q L L p x q B x x pW log τ w q B pv q Bt 1 n 1 1 1 1 L L B B B p q B B log τn B w1 B1 v1 (195) t1 s1 L t1 L L

p x p where we have used in pxq that rB1, V1s 0 and in p x q that V1 B{Bt1. L L So we must compute

B x B2 B nc W , w , B pv q 0 , (196) B 1 B B B 1 2 1 1 t1 L t1 s1 t1 L c 1 L Ep 1 1 q and so conclude that τn τn t 2 e2, s 2 e2,C B2 nc log τ B A log τ . (197) B B n 1 1 n 2 t1 s1 L c 1 The crucial points in this calculation were: B B r p s p x B1, V1 0 , V1 B , W1 B , (198) L L t1 s1 and indeed this is a model calculation, which shall be repeated over and over again. And so in the same fashion, conclude: B2 B2 B B log τn B2A1 log τn , B B log τn A2B1 log τn , (199) t1 s2 L s1 t2 L where we have used B{Bt1pnpn 1q{2q B{Bs1 pnpn 1q{2q 0.

3.7 Final equations

We have derived in Section 3.6, the following 52 Mark Adler

Relations on Locus L:

B B p q E E E E n n 1 log τn A1 log τn , log τn A2 log τn , Bt1 Bt2 2 B B p q E E E E n n 1 log τn B1 log τn , log τn B2 log τn , (200) Bs1 Bs2 2 B2 B2 E E E E log τn B2A1 log τn , log τn A2B1 log τn , Bt1Bs2 Bs1Bt2 B2 E E nc log τn B1A1 log τn 2 . Bt1Bs1 c 1 Remember from Section 3.4:

2–Toda relations

E 2 E B τ B {Bs Bt log τ log n 1 1 2 n , B E B2{B B E t1 τn1 t1 s1 log τn (201) E 2 E B τ B {Bt Bs log τ log n 1 1 2 n , B E B2{B B E s1 τn1 t1 s1 log τn Substitute the relations on L into the 2–Toda relations, which yields:

Pure boundary relations on the locus L

τ E E n1 A2B1 log τn A1 log E E 2 , τ A1B1 log τ nc{p1 c q n 1 n (202) E E τ B A log τ B log n 1 2 1 n . 1 E E {p 2q τn1 B1A1 log τn nc 1 c

Since A1B1 B1A, conclude that B A log τ E A B log τ E A 2 1 n B 2 1 n . (203) 1 E {p 2q 1 E {p 2q B1A1 log τn nc 1 c A1B1 log τn nc 1 c

R Notice that since τn is independent of ai and bi, we ﬁnd that E E R A1 log τn A1 log τn A1 log τn E τn p q A1 log R A1 log Pn E , (204) τn E Ñ p q and so (203) is true with log τn log Pn E , yielding the ﬁnal equation for FnpEq 1{n log PnpEq, and proving Theorem 3.1. Integrable Systems, Random Matrices and Random Processes 53 4 Dyson Brownian Motion and the Airy Process

4.1 Processes

The joint distribution for the Dyson process at 2–times deforms naturally to the 2–Toda integrable system, as it is described by a coupled Hermitian matrix integral, analyzed in the previous section. Taking limits of the Dyson process leads to the Airy and Sine processes. We describe the processes in this section in an elementary and intuitive fashion. A good reference for this discussion would be [33] and Dyson’s celebrated papers [30, 31] on Dyson diﬀusion. A random walk corresponds to a particle moving either left or right at 1 time n with probability p. If the particle is totally drunk, one may take p 2 . In that case, if Xn is its location after n steps, p q p 2q E Xn 0,E Xn n , (205) and in any case, this discrete process has no memory: P Xn1 j | pXn iq X parbitrary past eventq P pX1 j | X0 iq ,

1 i.e. it is Markovian. In the continuous version of this process (say? with p 2 ), rt{δs steps are taken in time t and each step is of magnitude δ, consistent with the scaling of (205). By the central limit theorem (CLT) for the binomial distribution, or in other words by Stirlings formula, it follows immediately that

2 epXXq {2t lim P pXt PpX,XdXq|X0 Xq ? dX :P pt, X,XqdX . (206) δÑ0 2πt

Note that P pt, X,Xq satisﬁes the (heat) diﬀusion equation

BP 1 B2P . (207) Bt 2 BX2 ? 1 The limiting motion where the particle moves δ with equal probability 2 in time δ, is in the limit, as δ Ñ 0, Brownian motion. The process isa scale invariant and so infinitesimally its fluctuations in t are no larger than ∆ptq and hence while the paths are continuous, they are nowhere differentiable (for almost all initial conditions.) We may also consider Brownian motion in n directions, all independent, and indeed, it was first observed in n 2 directions, under the microscope by Robert Brown, an English botanist, in 1828. In general, by independence,

¹n ° 1 npX X q2{2t{β P pt, X,Xq P pt, X ,X q e 1 i i , (208) i i pp q{ qn{2 1 2πt β hence 54 Mark Adler

BP 1 ¸n B2 P, (209) Bt 2β BX2 1 i where we have changed the variance and hence the diffusion constant from 1 Ñ β. In addition (going back to n 1), besides changing the rate of diffusion, we may also subject the diffusing particle located at X, to a harmonic force ρX, pointing toward the origin. Thus you have a drunken particle executing Brownian motion under the influence of a steady wind pushing him towards the origin — the Ornstein–Uhlenbeck (see [33]) process — where now the probability density P pt, X,Xq is given by the diffusion equation: BP 1 B2 B pρXq P. (210) Bt 2β BX2 BX

This can immediately be transformed to the case ρ 0, yielding pc eρtq

1 2 2 P pt, X,Xq ? epXcXq {pp1c q{ρβq , (211) 2πpp1 c2q{2ρβq1{2 which as ρ Ñ 0, p1 c2q{2ρ Ñ t, transforms to the old case. This process becomes stationary, i.e. the probabilities at a ﬁxed time do not change in t, if and only if the initial distribution of X is given by the limiting t Ñ 8 distribution of (211): 2 eρβX a dX, (212) π{ρβ and this is the only “normal Markovian process” with this property. 2 Finally, consider a Hermitian matrix B pBijq with n real quantities Bij undergoing n2 independent Ornstein–Uhlenbeck processes with ρ 1, but # β 1 for Bii (on diagonal) ,

β 2 for Bij (oﬀ diagonal) , and so the respective probability distribution Pii, Pij satisfy by (210): $ 'B B2 B ' Pii 1 p q & 2 Bii Pii , Bt 2 BB BBii ii (213) 'BP 1 B2 B %' ij pB q P , B B 2 B ij ij t 2 2 Bij Bij with solution, by (211) pc etq, given by: $ 1 p q2{p 2q ' Bii cBii 1 c &Piipt, Bii,Biiq ? a e , 2π p1 c2q{2 (214) ' 1 2 2 ' pBij cBij q {pp1c q{2q %Pij pt, Bij,Bij q ? a e . 2π p1 c2q{4 Integrable Systems, Random Matrices and Random Processes 55

By the independence of the processes the joint probability distribution is given by:

¹n ¹ 1 Z 2 2 P pt, B,Bq P P e TrpBcBq {p1c q , (215) ii ij p1 c2qn2{2 i1 1¤i, j¤n, ij

2 2 with Z p2πqn {22pn n{2q, which by (213) and (215) evolves by the Ornstein–Uhlenbeck process: BP ¸n 1 B2 B p1 δ q B P Bt 4 ij BB2 BB ij i,j1 ij ij (216) ¸n 1 B B 1 p1 δ q ΦpBq P pBq , 4 ij BB BB ΦpBq i,j1 ij ij with ΦpBq expp tr B2q. Note that the most general solution (215) to (216) is invariant under the unitary transformation

pB,Bq Ñ pUBU 1,UBU 1q , (217) which forces the actual process (216) to possess this unitary invariance and in fact (216) induces a random motion purely on the spectrum of B. This motion, discovered by Dyson in [30, 31], is called Dyson diﬀusion, and indeed the Ornstein–Uhlenbeck process Bptq Bijptq

given by (216) with solution (215), induces Dyson Brownian motion: n λ1ptq, . . . , λnptq P R on the eigenvalues of Bptq. The transition probability P pt, λ,¯ λq satisfies the following diffusion equation: a BP ¸n 1 B2 B B log Φpλq P B B 2 B B t 2 λi λi λi 1 (218) 1 ¸n B B 1 Φpλq P 2 Bλ Bλ Φpλq i1 i i with ¹n 2 p q 2 p q λi Φ λ ∆n λ e , 1 which is a Brownian motion, where instead of the particle at λi feeling only the harmonic restoring force λi, as in the Ornstein–Uhlenbeck process, it feels the full force a B log Φpλq ¸ 1 F pλq : λ , (219) i Bλ λ λ i i ji i j 56 Mark Adler which acts to keep the particles apart. In short, the Vandermonde in Φpλq creates n–repelling Brownian motions, while the exponential term keeps them from flying out to infinity. The equation (218) was shown by Dyson in [30, 31] by observing that Brownian motion with a force term F pFiq is, in general, completely characterized infinitesimally by the dynamics: 2 Epδλiq Fipλqδt, E pδλiq δt , (220) and so in particular (216) yields p q p q2 1 p q E δBij Bijδt, E δBij 2 1 δij δt . (221) Then by the unitary invariance (217) of the process (216), one may set at time t: Bptq diag λ1ptq, . . . , λnptq , and then using the perturbation formula:

¸ 2 2 pδBjiq pδBjiq δλ δB , i ii pλ λ q t ji i j 2 compute Epδλiq and E pδλiq by employing (221), immediately yielding (220) with Fipλq given by (219). Thus by the characterization of (218) by (220), we have veriﬁed Dysons result (218). Remember an Ornstein–Uhlenbeck process has a stationary measure precisely if we take for the initial measure the equilibrium measure at t Ñ 8. So consider our Ornstein–Uhlenbeck transition density (215) with t Ñ 8 stationary distribution: 2 Z1e tr B dB , and with this invariant measure as initial condition, one ﬁnds for the joint p q distribution pc e t2 t1 q

P pBpt1q P dB1,Bpt2q P dB2q

1 dB1dB2 1{p1c2q TrpB22cB B B2q Z e 1 1 2 2 , (222) p1 c2qn2{2

pti1tiq and similarly pci e q compute

P pBpt1q P dB1,...,Bptkq P dBkq ¹k 2 {p 2 q p q2 1 tr B1 1 1 ci1 Tr Bi ci1Bi1 Zk e e dB1, . . . , dBk i2 ¹k ° ¹n p {p 2 q 2{p 2qq n 2 1 1 1 ci1 ci 1 ci j1 λj,i Zk e dλj,i i1 j1 k¹1 2 p2ci{p1c qq λ`,i1λm,i detpe i q1¤`, ∆npλ1q∆npλkq, m¤n i1 λi pλ1,i, λ2,i, . . . , λn,iq , (223) Integrable Systems, Random Matrices and Random Processes 57 using the HCIZ formula (138). The distribution of the eigenvalues for GUE is expressible as a Fredholm determinant (9) involving the famous Hermite kernel (5) and Eynard and Mehta [32] showed that you have for the Dyson process an analogous extended Hermite kernel, speciﬁcally the matrix kernel [39]: $ ¸8 ' p q ' k ti tj p q p q ¥ &' e ϕnk x ϕnk y , ti tj , KH,n : k1 (224) titj ' ¸0 ' p q ' k tj ti p q p q %' e ϕnk x ϕnk y , ti tj , k8 with » x2{2 ϕipxqϕjpxq dx δij, ϕipxq pipxqe , R where $ p q & x2{2 Hk x e pkpxq, for k ¥ 0, with pkpxq ? , p q k{2 1{4 ϕk x % 2 k!π 0, for k 0 ; so pkpxq are the normalized Hermite polynomials. Then we have

H,E Prob (all Bptiq eigenvalues R Ei, 1 ¤ i ¤ m) detpI K q: H,Ep q p q H,np q p q Kij x, y IEi x Ktitj x, y IEj y , (225) the above being a Fredholm determinant with a matrix kernel.

Remark. In general such a Fredholm determinant is given by:

p p q q det I z Ktitj 1¤i,j¤m z1 » ¸8 ¸ ¹r1 ¹rm p qN p1q pmq 1 z dαi dαi R N1 °0¤ri¤N, 1 1 m r N 1 i pkq p`q p q ¤ ¤ det Ktkt` αi , αj 1 i rk, , ¤ ¤ 1 j r` 1¤k ,`¤m z1 where the N–fold integral above is taken over the range $ , ' 8 p1q ¤ ¤ p1q 8 / & α1 αr1 . R . , ' . / %8 pmq ¤ ¤ pmq 8- α1 αrm 58 Mark Adler

with integrand equal to the determinant of an N N matrix, with blocks given pkq p`q p q ¤ ¤ by the rk r` matrices Ktkt` αi , αj 1 i rk,. In particular, for m 2, we 1¤j¤r` have » ¸8 ¸ ¹r ¹s N 1 pzq ! ) dαi dβi 8 α ¤¤α 8 ¤ ¤ 1 r N 1 0 r,s N, 8 β1¤¤βs 8 1 1 rsN p p q p p q Kt1t1 αi, αj 1¤i, Kt1t2 αi, βj 1¤i¤r, det j¤r 1¤j¤s . (226) p p q p p q Kt2t1 βi, αj 1¤i¤s, Kt2t2 βi, βj 1¤i, 1¤j¤r j¤s z1 These processes have scaling limits corresponding to the bulk and edge scaling limits in the GUE. The Airy process is defined by rescaling in the extended Hermite kernel: ? u ? v τ x 2n ? , y 2n ? , t , (227) 2n1{6 2n1{6 n1{3 and the Sine process by rescaling in the extended Hermite kernel: uπ vπ τ x ? , y ? , t π2 . (228) 2n 2n 2n This amounts to following, in slow time, the eigenvalues at the edge and in the bulk, but with a microscope specified by the above rescalings. Then the extended kernels have well-defined limits as n Ñ 8: $ » 8 ' ' zptitj q & e Aipx zqAipy zq dz, ti ¥ tj , KA px, yq »0 (229) titj ' 0 ' zptj tiq % e Aipx zqAipy zq dz, ti tj , 8 $ » π ' 1 2 ' pz {2qptitj q & e cos zpx yq dz, ti ¥ tj , S π »0 Kt t 8 (230) i j ' 1 2 ' pz {2qptj tiq % e cos zpx yq dz, ti tj , π π with Ai the Airy function. Letting Aptq and Sptq denote the Airy and Sine processes, we define them below by

A,E ProbpAptiq R Ei, 1 ¤ i ¤ kq detpI K q , (231) S,E ProbpSptiq R Ei, 1 ¤ i ¤ kq detpI K q , where the determinants are matrix Fredholm determinants defined by the matrix kernels (229) and (230) in the same fashion as (225). The Airy process was first defined by Prähoferand Spohn in [46] and the Sine process was first defined by Tracy and Widom in [53]. Integrable Systems, Random Matrices and Random Processes 59

4.2 PDE’s and asymptotics for the processes

It turns out that the 2–time joint probabilities for all three processes, Dyson, Airy and Sine satisfy PDE’s, which moreover lead to long time t t2 t1 asymptotics in, for example, the Airy case. In this section we state the results of Adler and van Moerbeke [6], sketching the proofs in the next section. The ﬁrst result concerns the Dyson process:

Theorem 4.1 (Dyson process). Given t1 t2 and t t2 t1, the logarithm of the joint distribution for the Dyson Brownian motion λ1ptq, . . . , λnptq ,

Gnpt; a1, . . . a2r; b1, . . . , b2sq : log P pall λipt1q P E1, all λipt2q P E2q , satisﬁes a third-order nonlinear PDE in the boundary points of E1 and E2 and t, which takes on the simple form, setting c et,

B2A1Gn A2B1Gn A1 B1 . (232) B1A1Gn 2nc A1B1Gn 2nc

The sets E1 and E2 are the disjoint union of intervals ¤r ¤s E1 : ra2i1, a2is and E2 : rb2i1, b2is R , i1 i1 which specify the linear operators

¸2r B ¸2s B A c , 1 Ba Bb 1 j 1 j ¸2r B ¸2s B B c , 1 Ba Bb 1 j 1 j ¸2r B ¸2s B B A a c2 b p1 c2q c2 , 2 j Ba j Bb Bt 1 j 1 j ¸2r B ¸2s B B B c2 a b p1 c2q c2 . 2 j Ba j Bb Bt 1 j 1 j

The duality ai Ø bj reﬂects itself in the duality Ai Ø Bi. The next result concerns the Airy process:

Theorem 4.2 (Airy process). Given t1 t2 and t t2 t1, the joint distribution for the Airy process Aptq,

Gpt; u1, . . . u2r; v1, . . . , v2sq : log P pApt1q P E1,Apt2q P E2q , satisﬁes a third-order nonlinear PDE in the ui, vi and t, in terms of the 1 1 Wronskian tfpyq, gpyquy : f pyqgpyq fpyqg pyq, 60 Mark Adler 2 pLu LvqpLuEv LvEuq t pLu LvqLuLv G 1 2 2 2 tp q p q u 2 Lu Lv G, Lu Lv G Lu Lv . (233)

The sets E1 and E2 are the disjoint union of intervals ¤r ¤s E1 : ru2i1, u2is and E2 : rv2i1, v2is R , i1 i1 which specify the set of linear operators ¸2r B ¸2s B L : ,L : , u Bu v Bv 1 i 1 i ¸2r B B ¸2s B B E : u t ,E : v t . u i Bu Bt v i Bv Bt 1 i 1 i

The duality vi Ø vj reﬂects itself in the duality Lu Ø Lv, Eu Ø Ev.

Corollary 4.1. In the case of semi-inﬁnite intervals E1 and E2, the PDE for the Airy joint probability y x y x Hpt; x, yq : log P Apt q ¤ ,Apt q ¤ , 1 2 2 2

2 takes on the following simple form in x, y and t , with t t2 t1, also in terms of the Wronskian, " * B3H B B B2H B2H B2H B2H 2t t2 x 8 , . (234) B B B B B B 2 B 2 B B B 2 t x y x y x y x y y y Remark. Note for the solution Hpt; x, yq, y x y x lim Hpt; x, yq log F2 min , . t×0 2 2

The following theorem concerns the Sine process and uses the same sets and operators as Theorem 4.2:

Theorem 4.3 (Sine process). For t1 t2, and compact E1 and E2 R, the log of the joint probability for the Sine processes Siptq, p p p q P c p q P cq G t; u1, . . . u2r; v1, . . . , v2s : log P all Si t1 E1, all Si t2 E2 , satisﬁes

p2E L pE E 1qL qG L v u v u v u pL L q2G π2 u v p p q q 2EuLv Eu Ev 1 Lu G Lv 2 2 . (235) pLu Lvq G π Integrable Systems, Random Matrices and Random Processes 61

Corollary 4.2. In the case of a single interval, the logarithm of the joint probability for the Sine process,

Hpt; x, yq log P pSpt1q R rx1 x2, x1 x2s,Spt2q R ry1 y2, y1 y2sq satisﬁes

B p2EyB{Bx1 pEy Ex 1qB{By1qH 2 2 Bx1 pB{Bx1 B{By1q H π B p B{B p qB{B q 2Ex y1 Ex Ey 1 x1 H 2 2 . (236) By1 pB{Bx1 B{By1q H π Asymptotic consequences:

Pr¨ahoferand Spohn showed that the Airy process is a stationary process with continuous sample paths; thus the probability P pAptq ¤ uq is independent of t, and is given by the Tracy–Widom distribution » 8 2 P pAptq ¤ uq F2puq : exp pα uqq pαq dα , (237) u with qpαq the solution of the Painlev´e II equation, $ 3{2 &' ep2{3qα 2 ? , for α Õ 8 , q αq 2q3 with qpαq 2 πα1{4 (238) %'a α{2 , for a × 8 .

The PDE’s obtained above provide a very handy tool to compute large time asymptotics for these diﬀerent processes, with the disadvantage that one usually needs, for justiﬁcation, a nontrivial assumption concerning the interchange of sums and limits, which can be avoided upon directly using the Fredholm determinant formula for the joint probabilities (see Widom [57]) the latter method, however, tends to be quite tedious and quickly gets out of hand. We now state the following asymptotic result:

Theorem 4.4 (Large time asymptotics for the Airy process). For large t t2 t1, the joint probability admits the asymptotic series

P pApt q ¤ u, Apt q ¤ vq 1 2 F 1puqF 1pvq Φpu, vq Φpv, uq 1 F puqF pvq 2 2 O , (239) 2 2 t2 t4 t6 with the function q qpαq given by (238) and

p q Φ u, v » 8 2 » 8 2 » 8 2 1 2 2 2 1 2 1 2 : F2puqF2pvq q dα q dα q puq q pvq q dα 4 u v 4 2 v 62 Mark Adler » 8 » 8 dαp2pv αqq2 q12 q4q q2dα . (240) v u

Moreover, the covariance for large t t2 t1 behaves as 1 c E Apt qApt q E Apt q E Apt q , (241) 2 1 2 1 t2 t4 where ¼ c : 2 Φpu, vq du dv .

R2 Conjecture. The Airy process satisﬁes the nonexplosion condition for ﬁxed x:

lim P pAptq ¥ x z | Ap0q ¤ zq 0 . (242) zÑ8

4.3 Proof of the results

In this section we sketch the proofs of the results of the prior section, but all of these proofs are ultimately based on a fundamental theorem that we have proven in Section 3, which we now restate. Let M1,M2 P Hn, Hermitian n n matrices and consider the ensemble:

³ 2 2 1{2 TrpM M 2cM1M2q dM1dM2e 1 2 P pM ,M q S ³ S , (243) 1 2 1{2 TrpM 2M 22cM M q dM1dM2e 1 2 1 2 HnHn with ¹n ¹n 2 p q 2 p q dM1 ∆n x dxi dU1 , dM2 ∆n y dyi dU2 . 1 1 Given ¤r ¤s E E1 E2 ra2i1, a2is rb2i1, b2is , 1 1 deﬁne the boundary operators: 1 ¸r B ¸s B ¸r B B A˜ c , A˜ a , 1 c2 1 Ba Bb 2 j Ba Bc 1 j 1 j 1 j B˜1 A˜ , B˜2 A˜ . 1 2 aØb aØb

Note A˜1B˜1 B˜1A˜1. The following theorem was proven in Section 3:

Theorem 4.5. The statistics

Fnpc; a1, . . . , a2r; b1, . . . , b2sq : log PnpEq Integrable Systems, Random Matrices and Random Processes 63

: log P pall pM1–eigenvaluesq P E1, all pM2–eigenvaluesq P E2q , satisﬁes the third order nonlinear PDE: B˜2A˜1Fn A˜2B˜1Fn A˜1 B˜1 . (244) 2 2 B˜1A˜1Fn nc{p1 c q A˜1B˜1Fn nc{p1 c q Proof of Theorem 4.1.

Changing limits of integration in the integral Fn defined by the measure (243) to agree with the integral Gn defined by the measure (222), we find the function Gn of Theorem 4.1 is related to the function Fn of Theorem 4.5 by a trivial rescaling: a1 a2r Gnpt; a1, . . . , a2r; b1, . . . , b2sq Fn c; a ,..., a ; p1 c2q{2 p1 c2q{2

b b a 1 ,..., a 2s (245) p1 c2q{2 p1 c2q{2 and applying the chain rule to (244) using (245) leads to Theorem 4.1 immediately, upon clearing denominators. In order to prove the theorems concerning the Airy and Sine processes, we need a rigorous statement concerning the asymptotics of our Dyson, Airy and Sine kernels. To that end, letting $ , ? u &' x ÞÑ 2n 1 ? ./ t s 2n1{6 S : t ÞÑ , s ÞÑ , ? 1 ' 1{3 1{3 v / % n n y ÞÑ 2n 1 ? - 1{6 (246) " 2n* π2t π2s πu πv S2 : t ÞÑ , s ÞÑ , x ÞÑ ? , y ÞÑ ? , 2n 2n 2n 2n we have:

Proposition 4.1. Under the substitutions S1 and S2, the extended Hermite kernel tends with derivative,respectively, to the extended Airy and Sine kernel, when n Ñ 8, uniformly for u, v P compact subsets R:

H,np q A p q lim Kt,s x, y dy Kt,s u, v dv , nÑ8 S1 (247) H,np q pπ2{2qptsq S p q lim Kt,s x, y dy e Kt,s u, v dv . nÑ8 S2 Remark. The proof involves careful estimating and Riemann–Hilbert techniques and is found in [6]. 64 Mark Adler

Proof of Theorem 4.2.

Rescale in Theorem 4.1 ? ? ui vi τ ai 2n ? , bi 2n ? , t (248) 2n1{6 2n1{6 n1{3 and then from Proposition 4.1 it follows that, with derivatives that, τ 1 G , a; b Gpτ, u, vq O , k n1{6 . (249) n n1{3 k

We now do large n asymptotics on the operators Ai, Bi, setting L Lu Lv, E Eu Ev, with Lu,Lv,Eu,Ev deﬁned in Theorem 4.2; we ﬁnd: ? τ τ 2 τ 3 1 A 2k L L O , 1 k2 2k4 6k6 v k8 ? 2 3 τ τ τ 1 B1 2k L 2 4 6 Lu O 8 , k 2k 6k k 2τ 1 A 2k4 L L pE 1 4τ 2L q 2 k2 v 2k4 v (250) τ 4 2 1 6 Ev 1 τ Lv O 8 , k 3 k 4 2τ 1 p 2 q B2 2k L 2 Lu 4 E 1 4τ Lu k 2k τ 4 1 E 1 τ 2L O , k6 u 3 u k8 and consequently 1 ? B2A1 2 2k5 2 τ 1 2 L pL LuqL LpE 2q τ 4LupL Lvq LLvq k2 2k4 2 τ p q 1 p q τ p q 6 L Eu 2 Lv E 2 8LLu 18LuLv LLv k 2 6 1 O (251) k8 1 τ τ 2 1 τ 3 1 1 B A L2 L2 L2 L L L2 L L O . 2k2 1 1 k2 k4 2 u v k6 6 u v k8

Feeding these estimates and (249) into the relation (232) of Theorem 4.1, 2 multiplied by pB1A1Gn2ncq , which by the quotient rule becomes an identity involving Wronskians, we then ﬁnd ptf, guX gXf fXgq Integrable Systems, Random Matrices and Random Processes 65 " * 1 1 2 0 ? B A G , pB A G 2k6eτ{k q 5 2 1 n 2 1 1 n ? 2 2k 2k {p q " A1 2k * 1 1 2 ? A B G , pA B G 2k6eτ{k q 5 2 1 n 2 1 1 n ? 2 2k 2k {p q B1 2k 2τ p qp q 2p q 2 Lu Lv LuEv LvEu τ Lu Lv LuLv Gn k 1 2 2 2 1 tp q p q u Lu Lv Hn, Lu Lv Gn Lu Lv O 3 2 k 2τ p qp q 2p q 2 Lu Lv LuEv LvEu τ Lu Lv LuLv G k 1 2 2 2 1 tpL L qG, pL L q Gu O . 2 u v u v Lu Lv k3

In this calculation, we used the linearity of the Wronskian tX,Y uZ in its three arguments and the following commutation relations:

rLu,Eus Lu, rLu,Evs rLu,Lvs rLu, τs 0, rEu, τs τ ,

Ø t 2 u including their dual relations by u v; also we have L G, 1 LuLv tLpLu LvqG, 1uL. It is also useful to note that the two Wronskians in the ﬁrst expression are dual to each other by u Ø v. The point of the computation is to preserve the Wronskian structure up to the end. This proves Theorem 4.2, upon replacing τ Ñ t.

Proof of Corollary 4.1.

Equation (233) for the probability

Gpτ; u, vq : log P pApτ1q ¤ u, Apτ2q ¤ vq, τ τ2 τ1 , takes on the explicit form B B2 B2 B3G B2G B2G B2G τ G 2 u v τ 2 Bτ Bu2 Bv2 Bu2Bv Bv2 BuBv Bu2 B3G B2G B2G B2G 2 u v τ 2 Bv2Bu Bu2 BuBv Bv2 B3G B B3G B B B G. (252) Bu3 Bv Bv3 Bu Bu Bv

This equation enjoys an obvious u Ø v duality. Finally the change of variables in the statement of Corollary 4.1 leads to (234). The proof of Theorem 4.3 is done in the same spirit as that of Theorem 4.2 and Corollary 4.2 follows immediately by substitution in Theorem 4.3. Next, we need some preliminaries to prove Theorem 4.4. The ﬁrst being: 66 Mark Adler

Proposition 4.2. The following ratio of probabilities admits the asymptotic expansion for large t ¡ 0 in terms of functions fipu, vq, symmetric in u and v

P pAp0q ¤ u, Aptq ¤ vq ¸ f pu, vq 1 i , (253) P pAp0q ¤ uqP pAptq ¤ vq ti i¥1 from which it follows that

lim P pAp0q ¤ u, Aptq ¤ vq P pAp0q ¤ uqP pAptq ¤ vq F2puqF2pvq , tÑ8 this means that the Airy process decouples at 8.

The proof necessitates using the extended Airy kernel. Note, since the probabilities in (253) are symmetric in u and v, the coeﬃcients fi are symmetric as well. The last equality in the formula above follows from stationarity and (237).

Conjecture. The coeﬃcients fipu, vq have the property

lim fipu, vq 0 for ﬁxed v P R , (254) uÑ8 and

lim fipz, z xq 0 for ﬁxed x P R . (255) zÑ8 The justiﬁcation for this plausible conjecture will now follow. First, con- sidering the following conditional probability: p p q ¤ p q ¤ q p p q ¤ | p q ¤ q P A 0 u, A t v P A t v A 0 u p p q ¤ q PA 0 u ¸ f pu, vq F pvq 1 i , 2 ti i¥1 and letting v Ñ 8, we have automatically ¸ fipu, vq 1 lim P pAptq ¤ v | Ap0q ¤ uq lim F2pvq 1 vÑ8 vÑ8 ti i¥1 ¸ f pu, vq 1 lim i , vÑ8 ti i¥1 which would imply, assuming the interchange of the limit and the summation is valid,

lim fipu, vq 0 , (256) vÑ8 Integrable Systems, Random Matrices and Random Processes 67 and, by symmetry

lim fipu, vq 0 . uÑ8

To deal with (255) we assume the following nonexplosion condition for any ﬁxed t ¡ 0, x P R, namely, that the conditional probability satisﬁes

lim P pAptq ¥ x z | Ap0q ¤ zq 0 . zÑ8 Hence, the conditional probability satisﬁes, upon setting

v z x , u z , and using limzÑ8 F2pz xq 1, the following:

¸ f pz, z xq 1 lim P pAptq ¤ z x | Ap0q ¤ zq 1 lim i , zÑ8 zÑ8 ti i¥1 which, assuming the validity of the same interchange, implies that

lim fipz, z xq 0 for all i ¥ 1 . zÑ8 Proof of Theorem 4.4.

Putting the log of the expansion (253)

Gpt; u, vq log P pAp0q ¤ u, Aptq vq ¸ h pu, vq log F puqlog F pvq i (257) 2 2 ti i¥1 f pu, vq f pu, vqf 2pu, vq{2 log F puqlog F pvq 1 2 1 , 2 2 t t2 into (252) leads to: (i) a leading term of order t, given by

Lh1 0 , (258) where B B B2 L : . (259) Bu Bv BuBv The most general solution to (258) is given by

h1pu, vq r1puq r3pvq r2pu vq , with arbitrary functions r1, r2, r3. Hence, 68 Mark Adler h pu, vq P pAp0q ¤ u, Aptq ¤ vq F puqF pvq 1 1 2 2 t with h1pu, vq f1pu, vq as in (253). Applying (254)

r1puq r3p8q r2p8q 0 for all u P R , implying r1puq constant r1p8q , and similarly r3puq constant r3p8q .

Therefore, without loss of generality, we may absorb the constants r1p8q and r3p8q in the deﬁnition of r2pu vq. Hence, from (257),

f1pu, vq h1pu, vq r2pu vq and using (255), 0 lim f1pz, z xq r2pxq , zÑ8 implying that the h1pu, vq–term in the series (257) vanishes. (ii) One computes that the term h2pu, vq in the expansion (257) of Gpt; u, vq sastisﬁes B3g B2g B3g B2g Lh with gpuq : log F puq . (260) 2 Bu3 Bv2 Bv3 Bu2 2 This is the term of order t0, by putting the series (257) in (252). The most general solution to (260) is

1 1 h2pu, vq g puqg pvq r1puq r3pvq r2pu vq .

Then

P pAp0q ¤ u, Aptq ¤ vq eGpt;u,vq ¸ h pu, vq F puqF pvq exp i 2 2 ti i¥2 h pu, vq F puqF pvq 1 2 . (261) 2 2 t2

In view of the explicit formula for the distribution F2 (237) and the behavior of qpαq for α Õ 8, we have that 1 1 lim g puq lim log F2puq uÑ8 uÑ8 » 8 lim q2pαq dα 0 . Ñ8 u u Integrable Systems, Random Matrices and Random Processes 69

Hence

0 lim f2pu, vq lim h2pu, vq r1p8q r3pvq r2p8q , uÑ8 uÑ8 showing r3 and similarly r1 are constants. Therefore, by absorbing r1p8q and r3p8q into r2pu vq, we have

1 1 f2pu, vq h2pu, vq g puqg pvq r2pu vq .

Again, by the behavior of qpxq at 8 and 8, we have for large z ¡ 0,

» 8 » 8 g1pzqg1pz xq q2pαq dα q2pαq dα ¤ cz3{2e2z{3 . z zx Hence 0 lim f2pz, z xq r2pxq zÑ8 and so 1 1 f2pu, vq h2pu, vq g puqg pvq , yielding the 1{t2 term in the series (257), and so it goes. Finally, to prove (241), we compute from (239), after integration by parts and taking into account the boundary terms using (238): ¼ B2 E Ap0qAptq uv P pAp0q ¤ u, Aptq ¤ vq du dv BuBv R2 » 8 » 8 1p q 1p q uF2 u du vF2 v dv 8 » 8 » 1 8 8 F 1puq du F 1pvq dv t2 2 2 ¼8 8 1 Φpu, vq Φpv, uq du dv t4 R2 1 O t6 2 1 c 1 E Ap0q O , t2 t4 t6 ¼ ¼ where c : Φpu, vq Φpv, uq du dv 2 Φpu, vq du dv ,

R2 R2 thus ending the proof of Theorem 4.4. 70 Mark Adler 5 The Pearcey Distribution

5.1 GUE with an external source and Brownian motion

In this section we discuss the equivalence of GUE with an external source, introduced by Br´ezin–Hikami [27] and a conditional Brownian motion, following Aptkarev, Bleher and Kuijlaars [21].

Non-intersecting Brownian paths:

Consider n–non-intersecting Brownian paths with predetermined endpoints at t 0, 1, as speciﬁed in Figure 2.

Fig. 2.

By the Karlin–McGregor formula [42], the above situation has probability density 1 pnpt, x1, . . . , xnq det ppαi, xj, tq 1¤i, det ppxi, aj, 1 tq 1¤i, Zn j¤n j¤n ¹n 1 2{ p q { { xi t 1 t p 2aixj tq p 2aixj 1 tq 1 e det e 1¤i, det e 1¤i, , Z ¤ ¤ n 1 j n j n (262) with12 2 epxyq {t ppx, y, tq ? . (263) πt 13 For example, let all the particles start out at x 0, at t 0, with n1 particles ending up at a, n2 ending up at a at t 1, with n n1 n2.

12 Here Zn Znpa, αq. 13 Obviously, implicit in this example is a well-deﬁned limit as the endpoints come together. Integrable Systems, Random Matrices and Random Processes 71

Fig. 3.

Here ψ px q ¤ ¤ 1 i j 1 i n1, 1¤j¤n1n2 pn ,n pt, x1, . . . , xnq ∆n n pxq det , 1 2 1 2 p q ¤ ¤ Zn1,n2,a ψi xj 1 i n2, 1¤j¤n1n2 with 2 ψpxq xi1ex {tp1tq2ax{p1tq . i r s So setting E 1¤i¤r b2i1, b2i , we ﬁnd

a a P pt, bq : Prob pall xiptq Eq n1,n2 n1,n2 n1 left-most paths end up at a p q n2 right-most paths end up at a : Prob all xi t E and all start at 0, with all paths non-intersecting. ψ px q i j 1¤i¤n1, ³ ± ¤ ¤ n p q 1 j n1 n2 n 1 dxi∆n x det E ψ px q i j 1¤i¤n2, ¤ ¤ 1 j n1 n2 . (264) ψ px q i j 1¤i¤n1, ³ ± ¤ ¤ n p q 1 j n1 n2 n 1 dxi∆n x det R ψ px q i j 1¤i¤n2, 1¤j¤n1n2 Random matrix with external source:

Consider the ensemble, introduced by Br´ezin–Hikami [27], on nn Hermitian matrices Hn 72 Mark Adler 1 P M P pM,M dMq etrpV pMqAMq dM , (265) Zn with A diagpa1, . . . , anq . By HCIZ (see (138)), we ﬁnd » » ¹n 1 p q 1 p q q 2 p q V zi tr AUΛU P spec M E ∆n z e dzi e dU Z n n E 1 Upnq » aizj ¹n detre s1¤i, 1 p q j¤n 2 p q V zi 1 ∆n z e dzi p q p q Zn En ∆n z ∆n a » 1 ¹n 1 p q p q V zi r aizj s 2 ∆n z e dzi det e 1¤i, . (266) Z n ¤ n E 1 j n For example: consider the limiting case

¤r A diag looooooooomooooooooonpa, a, . . . , a, looooomooooona, a, . . . , aq ,E rb2i1, b2is , (267) i1 n2 n1 P pspecpMq Eq : P pa, bq n1,n2 ρ px q i j 1¤i¤n1, ³ ± ¤ ¤ n p q 1 j n1 n2 n 1 dxi∆n z det E ρ px q i j 1¤i¤n2, ¤ ¤ 1 j n1 n2 . (268) ρ px q i j 1¤i¤n1, ³ ± ¤ ¤ n p q 1 j n1 n2 n 1 dxi∆n z det R ρ px q i j 1¤i¤n2, 1¤j¤n1n2 where p q i1 V pzqaz ρi z z e , n n1 n2 . (269) Then we have by Aptkarev, Bleher, Kuijlaars [21]:

Non-intersecting Brownian motion ô GUE with external source c d a p q 2t 2 Pn1,n2 t, b Pn1,n2 a, b , (270) 1 t tp1 tq V pzqz2{2 so the two problems: non-intersecting Brownian motion and GUE with an external source, are equivalent! Integrable Systems, Random Matrices and Random Processes 73

5.2 MOPS and a Riemann–Hilbert problem

In this section, we introduce multiple orthogonal polynomials (MOPS), following Bleher and Kuijlaars [24]. This lead them to a determinal m–point correlation function for the GUE with external source, in terms of a “Christoffel– Darboux” kernel for the MOPS, as in the pure GUE case. In addition, they formulated a Riemann–Hilbert (RH) problem for the MOPS, analogous to that for classical orthogonal polynomials, thus enabling them to understand universal behavior for the MOPS and hence universal behavior for the GUE with external source (see [21, 25]). Let us first order the spectrum of A of (267) in some definite fashion, for example

Ordered spectrum of pAq pa, a, a, a, . . . , aq : pα1, α2, . . . , αnq .

For each k 0, 1, . . . , n, let k k1 k2, k1, k2 deﬁned as follows:

k1 : # times a appears in α1, . . . , ak , (271) k2 : # times a appears in α1, . . . , ak . We now define the 2 kinds of MOPS. MOP II: Define a unique monic kth degree polynomial pk pk1,k2 : » p q p q p q p q pk x pk1,k2 x : pk1,k2 x ρi x dx 0 R p q i1 V pxqax ¤ ¤ ¤ ¤ ρi x x e , 1 i k1 , 1 i k2 , (272) MOP I: Define unique polynomials q pxq, q pxq of respective k11,k2 k1,k21 degrees k1 1, k2 1:

qk1pxq : qk ,k pxq q pxqρ pxq q pxqρ pxq 1 2 k»1 1,k2 1 k1,k2 1 1 j : x qk1pxq dx δj,k1 , 0 ¤ j ¤ k 1 , (273) R which immediately yields:

Bi-orthonal polynomials: »

pjpxqqkpxq dx δj,k j, k 0, 1, . . . , n 1 . (274) R This leads to a Christoﬀel–Darboux kernel, as in (5):

n¸1 paq p q p q 1{2V pxq1{2V pyq p q p q Kn1,n2 x, y : Kn x, y : e pk x qk y , (275) 0 which is independent of the ad hoc ordering of the spectrum of A and which, due to bi-orthonality, has the usual reproducing properties: 74 Mark Adler » 8 » 8

Knpx, xq dx n , Knpx, yqKnpy, zq dy Knpx, zq . (276) 8 8

The joint probability density can be written in terms of Kn,

1 TrpV pΛqAΛq 1 e ∆npλq detrKnpλi, λjqs1¤i, , (277) Zn n! j¤n with Λ diagpλ1, . . . , λnq, yielding the m–point correlation function:

Rmpλ1, . . . , λmq detrKnpλi, λjqs1¤i, , (278) j¤m and we ﬁnd the usual Fredholm determinant formula: c P pspecpMq E q det I Knpx, yqIEpyq . (279)

Finally, we have a Riemann–Hilbert (RH) problem for the MOPS.

Riemann–Hilbert problem for MOPS:

MOP II: p q pn1,n2 z Cpn1,n2 Cpn1,n2 p q p q Y z : c1pn11,n2 z c1Cpn11,n2 c1Cpn11,n2 (280) p q c2pn1,n21 z c2Cpn1,n21 c2Cpn1,n21 with C Cauchy transforms: » p q p q p q 1 f s ρ1 s ds p q V pzqaz Cf z : , ρ1 z e . (281) 2πi R s z Then Y pzq satisﬁes the RH problem: 1. Y pzq analytic on CzR. 2. Jump condition for x P R: 1 ρpxq ρpxq 1 1 Ypxq Ypxq 0 1 0 . 0 0 1

3. Behavior as z Ñ 8 zn1 n2 1 Y pzq I O z n1 . (282) z z n2

MOP I: A dual RH problem for q and pY 1qT . k1,k2 Finally we have a Christoffel–Darboux type formula (see (5)) for the kernel paq Kn,npx, yq of (275) expressed in terms of the RH matrix (280): Integrable Systems, Random Matrices and Random Processes 75 { p 2 2q 1 e 1 4 x y Kpaq px, yq p0, eay, eayqY 1pyqY pxq 0 . (283) n,n 2πipx yq 0 Thus to understand the large n asymptotics of the GUE with external source, from (277), (278), and (279), it suffices to understand the asymptotics of paq Kn,npx, yq given by (275). Thus by (283) it suffices to understand the asymptotics of the solution Y pzq to the RH problem of (282), which is the subject of [21] and [25].

5.3 Results concerning universal behavior In this section we will first discuss universal behavior for the equivalent random matrix and Brownian motion models, leading to the Pearcey process. We will then give a PDE of Adler–van Moerbeke [7] governing this behavior, and finally a PDE for the n-time correlation function of this process, deriving the first PDE in the following sections. The following pictures illustrate the situation. Universal behavior: ? I. Brownian motion: 2n paths, a n

Fig. 4.

1 At t 2 the Brownian paths start to separate into 2 distinct groups. 76 Mark Adler ? 2 II. Random matrices: n1 n2 n, V pzq z {2, a : aˆ 2n. ? ? ? p q p aˆ 2np qq{ Density of eigenvalues: ρ x : limnÑ8 Kn,n 2nx, 2nx 2n

I.a ˆ 1

II.a ˆ 1

III.a ˆ ¡ 1

Fig. 5.

The 3 corresponding regimes, I, II and III, for the random matrix density of states ρpxq and thus the corresponding situation for the Brownian motion, are explained by the following theorem: Theorem 5.1 (Aptkarev, Bleher, Kuijlaars [21]). For the GUE with external source, the limiting mean density of states for aˆ ¡ 0 is: ? ? ? Kaˆ 2np 2nx, 2nxq 1 ρpxq : lim n,n | Im ξpxq| , (284) nÑ8 2n π with

ξpxq: ξ3 xξ2 paˆ2 1qξ xaˆ2 0 (Pastur’s equation [45]) , yielding the density of eigenvalues pictures. It is natural to look for universal behavior neara ˆ 1 by looking with a microscope about x ?0. Equivalently, thinking in terms of the 2n–Brownian 1 motions, one sets a n and about t 2 one looks with a microscope near x 0 to see the 2n–Brownian motions slowly separating into two distinct groups. Rescale as follows: 1 τ u ? t ? , x , a n . (285) 2 n n1{4 Remembering the equivalence between Brownian motion and the GUE with external source, namely (270), the Fredholm determinant formula (279), for the GUE with external source, yields: Integrable Systems, Random Matrices and Random Processes 77

a p p q cq p r Eq Probn1,n2 all xi t E det I Kn , d ? d d 2 2ta{p1tq 2 2 (286) Kr Epx, yq K x, y I pyq . n tp1tq n1,n2 tp1 tq tp1 tq E

Ñ 8 r Ep q Ñ 8 Universal behavior for n amounts to understanding Kn x, y for n , under the rescaling (285) and indeed we have the results:

Universal Pearcey limiting behavior:

Theorem 5.2 (Tracy–Widom [54]). Upon rescaling the Brownian kernel, we find the following limiting behavior, with derivates: d ? d d {p q 1{4 2 2ta 1 t 2 2 ? lim n Kn,n x, y a n, nÑ8 tp1 tq tp1 tq tp1 tq ? 1 { t 2 τ n, px,yqpu,vq{n1{4 P p q Kτ u, v , with the Pearcy kernel Kτ pu, vq of Brézin–Hikami [27] defined as follows:

ppxqq2pyq p1pxqq1pyq p2pxqqpyq τppxqqpyq K px, yq : τ x y » 8 ppx zqqpy zq dz , (287) 0 where (note ω eiπ{4)

» 8 1 4 2 ppxq : eu {4τu {2iux du , 2π »8 1 4 2 qpyq : eu {4τu {2uy du (288) 2π X » 8 ω 4 2 Im dueu {4iτu {2peωuy eωuyq π 0 satisfy the diﬀerential equations (adjoint to each other)

p3 τp1 xp 0 and q3 τq1 yq 0 .

The contour X is given by the ingoing rays from 8eiπ{4 to 0 and the outgoing rays from 0 to 8eiπ{4.

Theorem 5.2 allows us to define the Pearcey process Ppτq as the motion 1 of an infinite number of non-intersecting Brownian paths, near t 2 , upon taking a limit in (286), using the precise scaling of (285), to wit: 78 Mark Adler ? n 1{4 1 ?τ R p P q lim Probn,n all n xi E det I Kτ IE nÑ8 2 n : ProbpPpτq R Eq , (289) which defines for us the Pearcey process. Note the pathwise interpretation of Ppτq certainly needs to be resolved. The Pearcey distribution with the parameter τ can also be interpreted as the transitional probability for the Pearcey process. We now give a PDE for the distribution, which shall be derived in the following section:

Theorem 5.3 (Adler–van Moerbeke [7]). r r s For compact E i1 u2i1, u2i ,

F pτ; u1, . . . , u2rq : log ProbpPpτq R Eq (290) satisﬁes the following 4th order, 3rd degree PDE in τ and the ui: 1 3 3 2 1 2 pB q{pB qp q t B {B u 2 F τ B0 2 B1F 16 B 1 F τ, B1F B1 B1 2 B BF {Bτ 1 0 , (291) where ¸2r B ¸2r B B ,B u . (292) 1 Bu 0 i Bu 1 i 1 i It is natural to ask about the joint Pearcey distribution involving k–times, namely: ? n 1{4 1 ?τj R ¤ ¤ lim Probn,n all n xi Ej, 1 j k nÑ8 2 n ProbpPpτ q R E , 1 ¤ i ¤ kq j j P p q ¤ det I IEi Kτiτj IEj 1 i, , (293) j¤k where the above is a Fredholm determinant involving a matrix kernel, and the P extended Pearcey kernel of Tracy–Widom [54] Kτiτj , is given by » » i8 1 4{ 2{ 4{ 2{ ds dt KP px, yq e s 4 τj s 2 ys t 4 τit 2 xt , (294) τiτj 2 4π X i8 s t

P p q P p q with X the same contour of (288). We note Kττ x, y Kτ x, y of (287), the Br´ezin–Hikami Pearcey kernel. We then have the analogous theorem to Theorem 5.3 (which we will not prove here) namely:

Theorem 5.4 (Adler–van Moerbeke [12]). p q p q rj r j j s ¤ ¤ For compact Ej i1 u2i1u2i , 1 j k, Integrable Systems, Random Matrices and Random Processes 79

p1q p2q pkq F pτ1, τ2, . . . , τk; u , u , . . . , u q : log ProbpP pτjq R Ej, 1 ¤ j ¤ kq (295)

pjq satisﬁes the following 4th order, 3rd degree PDE in τi and u : D1X 0, with

X : (296) 2 r 2 1 2 p q p q t u 4 E1 D 1D 1 E 1F 2E0 D0 2 D1F 8 D 1E 1F, D1F D1 2 D1E1F where

¸k ¸k piq r piq Dj : Bjpu q , D1 : τiB1pu q , i1 i1

¸2ri ¸k p q B B B pupiqq : pu i qj1 ,E : τ j 1 . j ` piq j i B Bτi `1 u` i1

5.4 3-KP deformation of the random matrix problem

In this section we shall deform the measures (269) in the random matrix problem, for V pzq z2{2, so as to introduce 3-KP τ–functions into the picture, and using the bilinear identities, we will derive some useful 3-KP PDE for these τ–functions. The probability distribution for the GUE with external source was given by (268), to wit: P pspecpMq Eq » ρ pz q ¹n i j 1¤i¤n1, 1 ¤ ¤ p q p q 1 j n1 n2 : Pn1,n2 E dzi∆n z det where Z n ρ pzj q ¤ ¤ n E 1 i 1 i n2, 1¤j¤n1n2 p q i1 z2{2az ρi z z e . (297) p q Let us deform ρi z as follows: ° 2{ 2 8 p q k p q Ñ p q i 1 z 2 az βz k1 tk sk z ρi z ρˆi z : z e e , (298) yielding a deformation of the probability:

τ pt, s, s,Eq P pEq Ñ n1,n2 . (299) n1,n2 p q τn1,n2 t, s , s , R Where, by the same argument used to derive (140), p q p q τn1,n2 t, s , s ,E : det mn1,n2 t, s , s ,E , (300) with 80 Mark Adler $ , ' r s / & µij 1¤i¤n1, . ¤ ¤ m pt, s, s,Eq : 0 j n1 n2 1 , (301) n1,n2 ' r s / % µij 1¤i¤n2, - 0¤j¤n1n21 and » p q p q µij t, s , s : ρˆij z dz . E

We also need the identity pn n1 n2q

τn ,n pt, s , s ,Eq : det mn ,n pt, s , s ,Eq 1 2 »1 2 ¹n1 ° ¹n2 ° 1 8 i 8 i tix tiy ∆npx, yq e 1 j e 1 j n1!n2! En j 1 j 1 ¹n1 ° 2{ 2 8 i p q xj 2 axj βxj 1 si xj ∆n1 x e e dxj j 1 ¹n2 ° 2{ 2 8 i p q yj 2 ayj βyj 1 si yj ∆n2 y e e dyj . (302) j1

That the above is a 3-KP deformation is the content of the following theorem.

Theorem 5.5 (Adler–van Moerbeke–Vanhaecke [15]). Given the functions τn1,n2 as in (300), the wave matrices

Ψ pλ; t, s, sq n1,n2 p q p q ψp1q p1qn2 ψ 2 ψ 3 n1,n2 n11,n2 n1,n21 1 p q p q p q p qn2 1 2 p qn2 3 : 1 ψn 1,n ψn ,n 1 ψn 1,n 1 , 1 2 1 2 1 2 τn1,n2pt,s ,s q p q p q p q 1 n2 1 2 3 ψ p1q ψ ψ n1,n2 1 n1 1,n2 1 n1,n2 Ψ pλ; t, s, sq (303) n1,n2 p q p q p q ψ 1 p1qn2 1ψ 2 ψ 3 n1,n2 n11,n2 n1,n21 1 p q p q p q p qn2 1 1 2 p qn2 3 : 1 ψn 1,n ψn ,n 1 ψn 1,n 1 , 1 2 1 2 1 2 τn1,n2pt,s ,s q p q p q ψ 1 p1qn21ψ 2 ψp3q n1,n21 n11,n21 n1,n2 with wave functions ° p q p q 8 i 1 p q n1 n2 1 tiλ p r 1s q ψn1,n2 λ; t, s , s : λ e τn1,n2 t λ , s , s , ° p2q n 8 s λi 1 p q 1 1 i p r s q (304) ψn1,n2 λ; t, s , s : λ e τn1,n2 t, s λ , s , ° p q 8 i 3 p q n2 1 si λ p r 1sq ψn1,n2 λ; t, s , s : λ e τn1,n2 t, s , s λ , satisfy the bilinear identity, Integrable Systems, Random Matrices and Random Processes 81 ¾ Ψ Ψ T dλ 0 , @k k , ` , ` , @t, s, t, s , (305) k1,k2 `1,`2 1 2 1 2 8 of which the p1, 1q component spelled out is: ¾ ° 8p q i p r 1s q p r 1s q k1 k2 `1 `2 1 ti ti λ τk1,k2 t λ , s , s τ`1,`2 t λ , s , s λ e dλ 8 ¾ p qk2 `2 p r 1s q p r 1s q 1 τk11,k2 t, s λ , s τ`11,`2 t, s λ , s 8 ° ` k 2 8ps s qλi λ 1 1 e 1 i i dλ (306) ¾ p r 1sq p r 1sq τk1,k21 t, s , s λ τ`1,`21 t, s , s λ 8 ° ` k 2 8ps s qλi λ 2 2 e 1 i i dλ 0 .

Sketch of Proof :

The proof is via the MOPS of Section 5.2. We use the formal Cauchy transform, thinking of z as large: » » p q p q ¸ f s ρˆ1 s 1 i1 Cfpzq : ds : ρˆ psqs ds , (307) z s zi 1 R i¥1 which should be compared with the Cauchy transform of (281), which we used in the Riemann Hilbert problem involving MOPS, and let C0 denote the Cauchy transform with 1 instead ofρ ˆ1 . We now make the following identi- Ñ fication between the MOPS of (272), (273) defined with ρi ρî (and so dependent on t, s, s) and their Cauchy transforms and shifted τ–functions, namely:

1 τk ,k pt rλ s, s , s q p pλq λk1 k2 1 2 , k1,k2 p q τk1,k2 t, s , s p r 1s qp qk2 k 1 τk11,k2 t, s λ , s 1 Cp pλq λ 1 , k1,k2 p q τk1,k2 t, s , s p r 1sq k 1 τk1,k21 t, s , s λ Cp pλq λ 2 , (308) k1,k2 p q τk1,k2 t, s , s 1 τk ,k pt rλ s, s , s q C q pλq λ k1 k2 1 2 , 0 k1,k2 p q τk1,k2 t, s , s 1 k 1 τ pt, s rλ s, s qp1q 2 k11 k1 1,k2 q pλq λ , k1 1,k2 p q τk1,k2 t, s , s 1 τ pt, s , s rλ sq k21 k1,k2 1 q pλq λ , k1,k2 1 p q τk1,k2 t, s , s 82 Mark Adler wherep pλq wastheMOPof thesecondkind and q pλq q pλqρˆ k1,k2 k1,k2 k11,k2 1 q pλqρˆ was the MOP of the first kind. This in effect identifies all the k1,k21 1 elements in the RH matrix Y pλq given in (280) and pY 1qT , the latter which also satisfies a dual RH problem in terms of ratio’s of τ–functions; indeed, Ψ pλq without the exponenials is precisely Y pλq, etc. for Ψ . Then us- k1,k2 k1,k2 ing a self-evident formal residue identity, to wit: ¾ » » p q 1 p q g s p q p q p q p q f z dµ s dz f s g s dµ s , (309) 2πi R s z R 8 ° p q 8 i p q1 p q with f z 0 aiz , and designating f t, s , s : f t, s , s , we immediately conclude that ¾ ° 8p q i 1 1 ti ti λ p q e pk1,k2 C0q`1,`2 λ dλ 8 » ° 8p q i 1 ti ti λ p q p q e pk1,k2 λ, t, s , s q`1,`2 λ, t, s , s dλ »R ° 8pt t qλi e 1 i i p pλ, t, s , s qpq pλ, t, s , s qρˆ k1,k2 `11,`2 1 R q pλ, t, s , s qρˆ q dλ ¾ `1,`2 1 1 ° 1 ps s qλi Cp pλq q pλq e i i dλ k1,k2 `11,`2 8 ¾ ° 1 ps s qλi Cp pλq q pλq e i i dλ . (310) k1,k2 `1,`21 8

By (308), this is nothing but the bilinear identity (306). In fact, all the other entries of (305) are just (306) with its subscripts shifted. To say a quick word about (308), all that really goes into it is solving explicitly the linear systems deﬁning pk ,k , q and q , namely (272) and (273), and making use 1 2 k1 1,k°2 k1,k2 1 p 8 i{ q p q 1 of the identity exp 1 x i 1 x for x small in the formula (300) p q for τk1,k2 t, s , s . An immediate consequence of Theorem 5.5 is the following: p q Corollary 5.1. Given the above τ-functions τk1,k2 t, s , s , they satisfy the following bilinear identities14

° 8 i °8 14 1 tiz p q i With e : 0 si t z deﬁning the elementary Schur polynomials. Integrable Systems, Random Matrices and Random Processes 83

¸8 ° 8p B{B B{B B{B q p q prB q 1 ak tk bk sk ck sk s`1`2k1k2j1 2a sj t e j0 τ`1,`2 τk1,k2 8 ¸ ° r 8pa B{Bt b B{Bs c B{Bs q s p2bqs pB qe 1 k k k k k k k1 `1 1 j j s (311) j 0 p qk2 `2 τ`11,`2 τk11,k2 1 8 ¸ ° r 8pa B{Bt b B{Bs c B{Bs q p q pB q 1 k k k k k k sk2`21j 2c sj s e j0 τ`1,`21 τk1,k21 0 , with a, b, c P C8 arbitrary. Upon specializing, these identities imply PDE’s expressed in terms of Hi- rota’s symbol for j 1, 2 ... :

B2 r 2 sjpBtqτk 1,k τk 1,k τ log τk ,k , (312) 1 2 1 2 k1,k2 B B 1 2 s1 tj1 B2 r 2 sjpBs qτk 1,k τk 1,k τ log τk ,k , (313) 1 2 1 2 k1,k2 B B 1 2 t1 sj1 yielding

2 B log τ τ τ k1,k2 k1 1,k2 k1 1,k2 , (314) B B τ 2 t1 s1 k1,k2 B B2{B B τk11,k2 t2 s1 log τk1,k2 log , (315) B B2{B B t1 τk1 1,k2 t1 s1 log τk1,k2 B B2{B B τk11,k2 t1 s2 log τk1,k2 log , (316) B B2{B B s1 τk1 1,k2 t1 s1 log τk1,k2 Proof. Applying Lemma A.1 to the bilinear identity (306) immediately yields (311). Then Taylor expanding in a, b, c and setting in equation (311) all ai, bi, ci 0, except aj1, and also setting `1 k1 2, `2 k2, equation (311) becomes B2 r 2 a 2s pB qτ τ τ τ Opa q 0, j 1 j t k1 2,k2 k1,k2 B B k1 1,k2 k1 1,k2 j1 s1 tj1 and the coeﬃcient of aj1 must vanish identically, yielding equation (312) upon setting k1 Ñ k1 1. Setting in equation (311) all ai, bi, ci 0, except bj1, and `1 k1, `2 k2, the vanishing of the coeﬃcient of bj1 in equation (311) yields equation (313). Specializing equation (312) to j 0 and 1 respectively yields (since s1ptq t1 implies s1pB˜tq B{Bt1; also s0 1):

2 B log τ τ τ k1,k2 k1 1,k2 k1 1,k2 B B τ 2 t1 s1 k1,k2 84 Mark Adler and B2 1 B B log τ τ τ τ τ . B B k1,k2 τ 2 Bt k11,k2 k11,k2 k11,k2 Bt k1 1,k2 s1 t2 k1,k2 1 1 Upon dividing the second equation by the ﬁrst, we ﬁnd equation (315) and similarly equation (316) follows from equation (313).

5.5 Virasoro constraints for the integrable deformations

Given the Heisenberg and Virasoro operators, for m ¥ 1, k ¥ 0:

p1q B J pmqt kδ , m,k B m 0,m tm ¸ 2 ¸ ¸ p q 1 B B J 2 ptq 2 it it jt m,k B B i B i j (317) 2 ti tj ¥ tim ij m i 1 i j m m 1 B kpk 1q k pmqtm δm,0 , 2 Btm 2

we now state (explicitly exhibiting the dependence of τk1,k2 on β): p q Theorem 5.6. The integral τk1,k2 t, s , s ; β; E , given by (302) satisﬁes

k1,k2 ¥ Bmτk1,k2 Vm τk1,k2 for m 1 , (318) where Bm and Vm are diﬀerential operators:

¸2r B ¤2r m1 B b , for E rb , b s R (319) m i Bb 2i 1 2i 1 i 1 and

p2q p1q Vk1,k2 : tJ ptq pm 1qJ ptq m m,k1k2 m,k1k2 p q p q p q J 2 psq aJ 1 psq p1 2βqJ 1 psq m,k1 m1,k1 m2,k1 p q p q p q J 2 psq aJ 1 psq p1 2βqJ 1 psqu . (320) m,k2 m1,k2 m2,k2 Lemma 5.1. Setting

¹k1 ° ¹k2 ° 8 i 8 i tix tiy dIn ∆npx, yq e 1 j e 1 j j1 j 1 ¹k1 ° 2{ 2 8 i p q xj 2 axj βxj 1 si xj ∆k1 x e e dxj j1 Integrable Systems, Random Matrices and Random Processes 85 ¹k2 ° 2{ 2 8 i p q yj 2 ayj βyj 1 si yj ∆k2 y e e dyj , j1 the following variational formula holds for m ¥ 1: d x ÞÑ x εxm1 i i i k1,k2 p q dIn ÞÑ m1 Vm dIn . (321) dε yi yi εyi ε0 Proof. The variational formula (321) is an immediate consequence of applying the variational formula (184) separately to the three factors of dIn, and in addition± applying± formula (185) to the ﬁrst factor, to account for the fact k1 k2 that j1 dxj j1 dyj is missing from the ﬁrst factor. Proof of Theorem 5.6: Formula (318) follows immediately from formula (321) by taking into account the variation of BE under the change of coordinates.

From (317) and (320) and from the identity when acting on on τk1,k2 , B B B , (322) Btn Bsn Bsn compute that 1 ¸ B2 B2 B2 Vk1,k2 m B B B B B B 2 ti tj s s s s i j m i j i j ¸ B B B iti isi isi Btim Bs Bs i¥1 i m i m B B pk k q pmqt k pmqs 1 2 B m 1 B m tm sm B p q p 2 2q p q k2 m sm k1 k1k2 k2 δm,0 a k1 k2 δm1,0 Bsm p q B m m 1 p q tm sm sm B 2 tm2 B B a pm 1qps s q Bs Bs m 1 m 1 m1 m 1 B B 2β , m ¥ 1 . (323) B B sm2 sm2 p q The following identities, valid when acting on τk1,k2 t, s , s ; β, E , will also be used: B 1 B B B 1 B B , , Bs 2 Bt1 Ba Bs 2 Bt2 Bβ 1 2 (324) B 1 B B B 1 B B , . B B B B B B s1 2 t1 a s2 2 t2 β 86 Mark Adler

Corollary 5.2. The τ-function τ τ pt, s, s; β, Eq satisfies the follow- °k1,k2 2r m1B{B ing differential identities, with Bm 1 bi bi. B B B1τ V1τ v1τ : B 2β B τ t1 a ¸ B B B iti isi isi τ Bt B B i¥2 i 1 si1 si1 papk k q k s k s pk k qt qτ , 2 1 11 2 1 1 2 1 1 B B B B τ W τ w τ : β τ 2 1 Ba 1 1 Bs Ba 1 ¸ B B B 1 iti isi isi τ 2 Bti1 Bs Bs i¥2 i 1 i 1 a 1 pk k q pk k qt k s k s τ 2 1 2 2 1 2 1 1 1 2 1 (325) B B B τ τ B0 aB τ V2τ v2τ : B 2β B a t2 β ¸ B B B it is is τ i Bt i B i B i¥1 i si si pk2 k2 k k qτ 1 2 1 2 1 B B Bτ Bτ B a τ W τ w τ : β 2 0 Ba Bβ 2 2 Bs Bβ 2 1 ¸ B B B it is is τ 2 i Bt i B i B i¥1 i si si 1 pk2 k2 k k qτ 2 1 2 1 2 where V1, W1, V2, W2 are first order operators and v1, w1, v2, w2 are functions, acting as multiplicative operators.

Corollary 5.3. On the locus L : tt s s 0, β 0u, the function p q f log τk1,k2 t, s , s ; β, E satisﬁes the following diﬀerential identities: B f p q B B1f a k1 k2 , t1 Bf 1 B a B f pk k q , Bs 2 1 Ba 2 2 1 1 B B f 2 2 B B0 aB f k1 k1k2 k2 , t2 a Bf 1 B B 1 B a f pk2 k2 k k q , (326) B 0 B B 1 2 1 2 s2 2 a β 2 Integrable Systems, Random Matrices and Random Processes 87 B2f B 2 B B f 2k , Bt Bs 1 Ba 1 1 1 1 B2f B B Bf 2 a B 1 B f 2 2apk k q , (327) Bt Bs Ba Bβ 0 1 Ba 1 2 1 2 B2f B B 2 B a aB f B pB 1qf 2apk k q . B B B 0 B 1 1 0 1 2 t2 s1 a a Proof. Upon dividing equations (325) by τ and restricting to the locus L, equations (326) follow immediately.

Remembering f log τ and setting 1 B A : B , B : B , 1 1 1 2 1 Ba (328) B 1 B B A : B a , B : B a , 2 0 Ba 2 2 0 Ba Bβ we may recast (325) as (compare with (193))

Akf Vkf vk, Bkf Wkf wk, k 1, 2 , (329) where (compare with (191)) we note that

B B V , W , k 1, 2 . (330) k B k B L tk L sk To prove (327) we will copy the argument of Section 3.6 (see (195)). Indeed, compute p q p q B1A1f L B1 V1f v1 B1V1f B1 v1 L L pq V1B1f B1pv1q L L p q B pW f w q B pv q Bt 1 1 1 1 1 L L B B Bw1 f B pv q , (331) B B B 1 1 t1 s L t1 L 1 L where we used in pq that rB1, V1s 0 and in p q that V1 B{Bt1, and L L so from (331) we must compute

2 B B Bw1 1 1 W , pk k q , B pv q pk k q . (332) B 1 B B B 1 2 1 1 1 2 t1 L t1 s1 t1 L 2 L 2 Now from (331) and (332), we ﬁnd 88 Mark Adler

B2f 1 1 B2f B A f pk k q pk k q k , 1 1 B B 1 2 1 2 B B 1 L t1 s1 2 2 t2 s1 L and so B2f 1 B B A f k B B f k , B B 1 1 1 1 1 B 1 t1 s1 L 2 a L L which is just the ﬁrst equation in (327). The crucial point in the calculation being (330) and rB1, V1s 0. The other two formulas in (327) are done L in precisely the same fashion, using the crucial facts (330) and rB2, V1s L rA2, W1s 0 and the analogs of (332). L

5.6 A PDE for the Gaussian ensemble with external source and the Pearcey PDE

From now on set: k1 k2 : k and restrict to the locus L. From (315) and (316) we have the

3-KP relations: N N B B2f B2f B B2f B2f g , g , (333) B B B B B B B B B B t1 t2 s1 t1 s1 s1 t1 s2 t1 s1 with f : log τk,k , g : logpτk1,k{τk1,kq , while from (326), we ﬁnd

Virasoro relations on L: B B B g g 1 B1g 2a , B1 g a . (334) Bt1 Bs 2 Ba B {B B {B Eliminating g t1, g s1 from (333) using (334) and then further elimi- 1 p B{B q nating A1g : Bg and B1g : 2 B1 a g using A1B1g B1A1g yields B2{B B B2{B B q t2 s1 2 t1 s2 f B 1 B2f{Bt Bs 1 1 B pB2{Bt Bs 2a B2{Bt Bsqf 2 1 1 1 , (335) B B2 {B B a f t1 s1 while from (327) we ﬁnd the Integrable Systems, Random Matrices and Random Processes 89

Virasoro relations on L: 2 B B 2 f B B f 2k : F , Bt Bs 1 Ba 1 1 1 2 2 B B Bf 2 2 f H 2B , (336) Bt Bs Bt Bs 1 1 BB 2 1 1 2 B2 B2 2 2a f H , B B B B 2 t2 s1 t1 s2 where the precise formulas for Hi will be given later. Substituting (336) into (335) and clearing the denominator yields15 " * " * " * Bf 1 1 B ,F H , F H , F , (337) 1 Bβ 1 2 2 2 B1 B1 B{Ba and by the involution: a Ñ a, β Ñ β, which by (302), clearly ﬁxes f p q log τk,k on t s s 0, we ﬁnd Hi Hi|aÑa " * " * " * Bf 1 1 B ,F H , F H , F . (338) 1 Bβ 1 2 2 2 B1 B1 B{Ba

These 2 relations (337) and (338) yield a linear system for: B B f 2 f B , B . 1 Bβ 1 Bβ Solving the system yields: B B f 2 f B R , B R , 1 Bβ 1 1 Bβ 2 and so B1R1pfq R2pfq , f log τk,kp0, 0, 0,Eq . β0 Since τk,kp0, 0, 0,Eq P pspecpMq Eq , τ p0, 0, 0, R k,k β0 with k¹1 2 k2 k ka2 k2 τk,kp0, 0, 0, Rq j! 2 p2πq e a , β0 0 we ﬁnd the following theorem:

15 tf, guX : gXf fXg. 90 Mark Adler

Theorem 5.7 (Adler–van Moerbeke [7]). hkkkkkikkkkkjk hkkkikkkjk rr s p q For E 1 b2i1, b2i , A diag a, . . . , a, a, . . . , a , ³ p 1 2 q eTr 2 M AM dM H2kpEq P pa; b1, . . . , b2rq ³ (339) p 1 2 q eTr 2 M AM dM H2kpRq satisﬁes a nonlinear 4th order PDE in a, b1, . . . br:

pF B1G F B1G qpF B1F F B1F q p qp 2 2 q F G F G F B1F F B1F 0 , (340) where ¸2r B ¸2r B B1 , B0 bi , Bbi Bbi 1 1 B F :2kB B log P, 1 Ba 1 G :tH ,F u tH ,F uB{B , 1 B1 2 a B B B 2 k H1 : B B0 aB aB1 4B log P B0B1 log P 4ak 4 , (341) a a a a B B 2 H : B a aB log P p2aB B B 2B q log P, 2 Ba 0 Ba 1 1 0 1 1 16 F F and G G . aÑa aÑa

We now show how Theorem 5.7 implies Theorem 5.3. Indeed, remember 1 our picture of 2k Brownian paths diverging at t 2 . Also, remembering the equivalence (270) between GUE with external source and the above Brownian motion, and (286), we ﬁnd

a p q a p p q q Pk,k t; b1, . . . , b2r : Probk,k all xi t E c d 2t 2 P a; pb , . . . , b q (342) 1 t tp1 tq 1 r p r Ec q det I K2k , where the function P p; q is that of (339). Letting the number of particles 2k Ñ 8 and looking about the location 1 x 0 at time t 2 with a microscope, and slowing down time as follows:

16 Note P pa; b1, . . . b2rq P pa; b1, . . . , brq. Integrable Systems, Random Matrices and Random Processes 91

Fig. 6.

1 1 k , a , b u z , t 1 τz2 , z Ñ 0 , (343) z4 z2 i i 2 which is just the Pearcey scaling (285), we ﬁnd by Theorem 5.2 and (342), that ¤r 1{z2 1 2 Prob all x τz P rzu , zu s p1{z4,1{z4q i 2 2i 1 2i 1 c d 2 2 P a; pb , . . . , b q 1 t tp1 tq 1 2r 1 { 2 2 a 1 z ,bi uiz,t 2 τz (344) p P q p q det I Kτ IErc O z : Qpτ; u1, . . . , u2rq Opzq , P r rr s where Kτ is the Pearcey kernel (287) and E 1 u2i1, u2i . Taking account of c d a 2t 2 P pt; b1, . . . , b2rq P a; pb1, . . . , br k,k 1 t tp1 tq (345)

: P pA, B1,...,B2rq , where the function P p, q is that of (339), and the scaling (343) of the Pearcey process, we subject the equation (340) to both the change of coordinates involved in the equation (345) and the Pearcey scaling (343) simultaneously: 0 tpF B1G F B1G qpF B1F F B1F q 2 2 d pF G F G qpF B F F B F qu ? 1 1 1 τz2 A 2 2 , z2 1 τz2 2? ?uiz 2 Bi 1 τ2z4 4 92 Mark Adler 1 1 PDE in τ and u for log P a pt; b , . . . , b q O z17 k,k 1 2r z15 scaling 1 1 same PDE for log Qpτ, u , . . . , u q O ; z17 1 2r z16 the ﬁrst step is accomplished by the chain rule and the latter step by (344), yielding Theorem 5.3 for F log Q.

A Hirota Symbol Residue Identity

Lemma A.1. We have the following formal residue identity ¾ ° 1 1 1 1 1 2 1 2 2 8pt1 t2qzi r fpt rz s, s , u qgpt rz s, s , u qe 1 i i z dz 2πi 8 ¸ °8 pa` B{Bt`b` B{Bs`c` B{Bu`q sj1rp2aqsjpB˜tqe 1 g f , (A.346) j¥0 where t1 t a , s1 s b , u1 u c , (A.347) t2 t a , s2 s b , u2 u c , ¸8 B B B °8 r 1 1 t zi i B , , ,... , e 1 i s ptqz , (A.348) t Bt 2 Bt 3 Bt i 1 2 3 0 and the Hirota symbol ppBt, Bs, Buqg f 1 1 1 1 1 1 : ppBt1 , Bs1 , Bu1 qgpt t , s s , u u qfpt t , s s , u u q . (A.349) t10, s10, u10 Proof. By definition, ¾ i¸8 1 i a z dz a , (A.350) 2πi i 1 8 i8 and so by Tayler’s Theorem, following [28] compute: ¾ ° 1 1 1 1 2 1 2 2 8pt1 t2qzi r dz fpt rz s, s , u qgpt rz s, s , u qe 1 i i z 2πi 8 ¾ ° 1 1 2 8 a zi r dz fpt a rz s, s b, u cqgpt a rz s, s b, u cqe 1 i z 2πi 8 Integrable Systems, Random Matrices and Random Processes 93 ¾ ° ° 8p1{iziqB{Ba 2 8 a zi r dz e 1 i fpt a, s b, u cqgpt a, s b, u cqe 1 i z 2πi 8 ¾ 8 8 ¸ ¸ dz zjs pB˜ qfpt a, s b, u cqgpt a, s b, u cq z`rs p2aq j a ` 2πi 8 1 `0 (picking out the residue term) ¸8 sj1rp2aqsjpBãqfpt a, s b, u cqgpt a, s b, u cq j0 8 ¸ °8 1 1 1 pa` B{Bt b` B{Bs c` B{Bu q 1 1 1 sj1rp2aqsjpBãqe 1 ` ` ` fpt t , s s , u u q j0 gpt t1, s s1, u u1q at t1 s1 u1 0 8 ¸ °8 1 1 1 pa` B{Bt b` B{Bs c` B{Bu q sj1rp2aqsjpB˜t1 qe 1 ` ` ` j0 gpt t1, s s1, u u1qfpt t1, s s1, u u1q at t1 s1 u1 0

8 ¸ °8 pa` B{Bt`b` B{Bs`c` B{Bu`q sj1rp2aqsjpB˜tqe 1 gptq fptq , j0 completing the proof.

Proof of (28):

To deduce (28) from (27), observe that since t, t1 are arbitrary in (27), when we make the change of coordinates (A.347), a becomes arbitrary and we then apply Lemma A.1, with s and u absent, r 0 and f g τ, to deduce (28).

Proof of (166):

To deduce (167) from (166), apply Lemma A.1 to the l.h.s. of (166), setting 1 f τn, g τm1, r n m 1, where no u, u is present, and when we make the change of coordinates (A.347), since t, t1, s, s1 are arbitrary, so is a and b, while in the r.h.s. of (166), we ﬁrst need to make the change of coordinates z Ñ z1, so znm1 dz Ñ zmn1 dz (taking account of the switch in orientation) and then we apply Lemma A.1, with f τn1, g τm, r m n 1, and so deduce (167) from (166).

References

1. M. Adler, P. van Moerbeke: Adv. Math. 108, 140 (1994) 2. M. Adler, P. van Moerbeke: Comm. Pure Appl. Math. 54, 153 (2001) 3. M. Adler, P. van Moerbeke: Comm. Pure Appl. Math. 237, 397 (2003) 94 Mark Adler

4. M. Adler, P. van Moerbeke: Ann. of Math. (2) 149, 921 (1999) 5. M. Adler, P. van Moerbeke: Ann. of Math. (2) 153, 149 (2001) 6. M. Adler, P. van Moerbeke: Ann. Probab. 33, 1326 (2005) 7. M. Adler, P. van Moerbeke: Math-pr/0509047 (2005) 8. M. Adler, P. van Moerbeke: Duke Math. J. 80, 863 (1995) 9. M. Adler, P. van Moerbeke: Comm. Pure Appl. Math. 50, 241 (1997) 10. M. Adler, P. van Moerbeke: Comm. Math. Phys. 203, 185 (1999) 11. M. Adler, P. van Moerbeke: Duke Math. J. 112, 1 (2002) 12. M. Adler, P. van Moerbeke: PDE’s for n time correlation of the Pearcey process and the n coupled Gaussian ensemble with external source (Preprint 2006) 13. M. Adler, P. van Moerbeke, P. Vanhaecke: Math-ph/0601020 (2006) 14. M. Adler, P. van Moerbeke, P. Vanhaecke: Algebraic Integrability, Painlevé Geometry and Lie Algebras (Springer 2004) 15. M. Adler, P. van Moerbeke, P. Vanhaecke: Moment matrices and multi- component KP, with applications to random matrix theory (Preprint 2006) 16. M. Adler, T. Shiota, P. van Moerbeke: Duke Math. J. 94, 379 (1998) 17. M. Adler, T. Shiota, P. van Moerbeke: Comm. Math. Phys. 171, 547 (1995) 18. M. Adler, T. Shiota, P. van Moerbeke: Phys. Lett. A 208, 67 (1995) 19. M. Abramowitz, I. Stegun: Handbook of Mathematical Functions (Dover 1972) 20. D. Aldous, P. Diaconis: Bull. Amer. Math. Soc. (N.S.) 36, 413 (1999) 21. A. Aptekarev, P. Bleher, A. Kuijlaars: Math-ph/0408041 (2004) 22. J. Baik, P. Deift, K. Johansson: J. Amer. Math. Soc. 12, 1119 (1999) 23. D. Bessis, Cl. Itzykson, J.-B. Zuber: Adv. in Appl. Math. 1, 109 (1980) 24. P. Bleher, A. Kuijlaars: Internat. Math. Res. Notices 3, 109 (2004) 25. P. Bleher, A. Kuijlaars: Comm. Math. Phys. 252, 43 (2004) 26. A. Borodin: Duke Math. J. 117, 489 (2003) 27. E. Brézin,S. Hikami: Phys. Rev. E 58, 7176 (1998) 28. E. Date, M. Jimbo, M. Kashiwara T. Miwa: Transformation groups for soliton equations. In: Nonlinear Integrable Systems—Classical Theory and Quantum Theory (World Sci. Publishing, Singapore, 1983) pp 39–119 29. P. Deift: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. In. Courant Lecture Notes in Mathematics, vol 3 (Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 1999) 30. F.J. Dyson: J. Math. Phys. 3, 1191 (1962) 31. F.J. Dyson: J. Math. Phys. 3, 140 (1962); 157 (1962); 166 (1962) 32. B. Eynard, M.L. Mehta: J. Phys. A 31, 4449 (1998) 33. W. Feller: An Introduction to Probability Theory and Its Applications, vol. I–II (Wiley 1966) 34. M. Gaudin: Nuclear Phys. 25, 447 (1961) 35. S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces (Aca- demic Press, Boston 1978) 36. M. Jimbo, T. Miwa, Y. Mori, M. Sato: Proc. Japan Acad. Ser. A Math. Sci. 55, 317 (1979) 37. M. Jimbo, T. Miwa, Y. Mori, M. Sato: Phys. D 1, 80 (1980) 38. K. Johansson: Duke Math. J. 91, 151 (1998) 39. K. Johansson: Math-pr/0306216 (2003) 40. V.G. Kac: Infinite-Dimensional Lie Algebras 3rd ed (Academic Press, Boston 1995) Integrable Systems, Random Matrices and Random Processes 95

41. V.G. Kac, A.K. Raina: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. In: Advanced Series in Mathematical Physics, vol 2 (World Scientific Publishing Co., Inc., Teaneck, NJ, 1987) 42. S. Karlin, J. McGregor: Pacific J. Math. 9, 1141 (1959) 43. E.M. McMillan: A problem in the stability of periodic systems. In: Topics in Modern Physics, A Tribute to E.U. Condon, vol 424, ed by W.E. Britten, H. Odabasi (Colorado Ass. Univ. Press, Boulder 1971) pp 219–244 44. M.L. Mehta: Random Matrices, 2nd edn (Academic Press, Boston 1991) 45. L.A. Pastur: Teoret. Mat. Fiz. 10, 102 (1972) 46. M. M. Prähofer,H. Spohn: J. Stat. Phys. 108, 1071 (2002) 47. E.M. Rains: Topics in probability on compact Lie groups. Ph.D. Thesis, Harvard University, Cambridge (1995) 48. C.A. Tracy, H. Widom: Introduction to random matrices. In: Geometric and Quantum Aspects of Integrable Systems, vol 424, ed by G.F. Helminck (Lecture Notes in Phys., Springer, Berlin 1993) pp 103–130 49. C.A. Tracy, H. Widom: Commun. Math. Phys. 159, 151 (1994) 50. C.A. Tracy, H. Widom: Commun. Math. Phys. 161, 289 (1994) 51. C.A. Tracy, H. Widom: Comm. Math. Phys. 207, 665 (1999) 52. C.A. Tracy, H. Widom: Probab. Theory Related Fields 119, 350 (2001) 53. C.A. Tracy, H. Widom: Math-pr/0309082 (2003) 54. C.A. Tracy, H. Widom: Math-pr/0412005 (2004) 55. K. Ueno, K. Takasaki: Adv. Stud. Pure Math. 4, 1 (1984) 56. P. Van Moerbeke: Integrable foundations of string theory. In: Lectures on Inte- grable Systems, ed by O. Babelon, P. Cartier, Y. Kosmann–Schwarzbach (World Sci. Publishing, River Edge, NJ, 1994 ) pp 163–267 57. H. Widom: Math-pr/0308157 (2003) 58. E.P. Wigner: Proc. Camb. Phil. Soc. 47, 790 (1951)

Integrable Systems, Random Matrices and Random Processes 97