Markov operators, classical orthogonal polynomial ensembles, and random matrices

Markov operators, classical orthogonal polynomial ensembles, and random matrices M. Ledoux, Institut de Mathématiques de Toulouse, France 5ecm Amsterdam, July 2008

recent study of random matrix and random growth models

recent study of random matrix and random growth models new asymptotics common, non central, rate (mean) 1/3 universal limiting Tracy-Widom distribution

recent study of random matrix and random growth models new asymptotics common, non central, rate (mean) 1/3 universal limiting Tracy-Widom distribution random matrices, longest increasing subsequence, random growth models, last passage percolation... P. Forrester, C. Tracy, H. Widom, J. Baik, P. Deift, K. Johansson

invariant random matrix models

invariant random matrix models P(dX ) = 1 Z exp ( Tr ( v(x ) )) dx, v : R R X = X N real symmetric or Hermitian N N matrix

invariant random matrix models P(dX ) = 1 Z exp ( Tr ( v(x ) )) dx, v : R R X = X N real symmetric or Hermitian N N matrix v(x) = β x 2 /4 Gaussian Orthogonal (β = 1) Unitary (β = 2) Ensembles

invariant random matrix models P(dX ) = 1 Z exp ( Tr ( v(x ) )) dx, v : R R X = X N real symmetric or Hermitian N N matrix v(x) = β x 2 /4 Gaussian Orthogonal (β = 1) Unitary (β = 2) Ensembles joint law of the eigenvalues (λ N 1 λn N ) of X N

invariant random matrix models P(dX ) = 1 Z exp ( Tr ( v(x ) )) dx, v : R R X = X N real symmetric or Hermitian N N matrix v(x) = β x 2 /4 Gaussian Orthogonal (β = 1) Unitary (β = 2) Ensembles joint law of the eigenvalues (λ N 1 λn N ) of X N P N (dx) = 1 Z N (x) β N i=1 dµ(x i ), x = (x 1,..., x N ) R N N (x) = i<j (x i x j ), dµ(x) = e v(x) dx

random growth models directed last passage percolation

random growth models directed last passage percolation w ij, 1 i, j N, iid geometric random variables

random growth models directed last passage percolation w ij, 1 i, j N, iid geometric random variables W N = max π (i,j) π w ij π up/right paths from (1, 1) to (N, N)

random growth models directed last passage percolation w ij, 1 i, j N, iid geometric random variables W N = max π (i,j) π w ij π up/right paths from (1, 1) to (N, N) K. Johansson (2000)

random growth models directed last passage percolation w ij, 1 i, j N, iid geometric random variables W N = max π (i,j) π w ij π up/right paths from (1, 1) to (N, N) P ( W N t) = K. Johansson (2000) {max x i t+n 1} 1 Z N(x) 2 N dµ(x i ) i=1

random growth models directed last passage percolation w ij, 1 i, j N, iid geometric random variables W N = max π (i,j) π w ij π up/right paths from (1, 1) to (N, N) P ( W N t) = K. Johansson (2000) {max x i t+n 1} 1 Z N(x) 2 N dµ(x i ) i=1 µ({x}) = (1 q)q x, x N geometric distribution

law of (λ N 1,..., λ N N ) : PN (dx) = 1 Z N (x) β N i=1 dµ(x i )

law of (λ N 1,..., λ N N ) : PN (dx) = 1 Z N (x) β global regime : spectral measure N i=1 dµ(x i )

law of (λ N 1,..., λ N N ) : PN (dx) = 1 Z N (x) β global regime : spectral measure N i=1 dµ(x i ) large deviation asymptotics (every β)

law of (λ N 1,..., λ N N ) : PN (dx) = 1 Z N (x) β global regime : spectral measure N i=1 dµ(x i ) P N (dx) = 1 Z exp ( N v(x i ) + β i<j i=1 ) log x i x j dx

law of (λ N 1,..., λ N N ) : PN (dx) = 1 Z N (x) β global regime : spectral measure N i=1 dµ(x i ) P N (dx) = 1 Z exp ( N v(x i ) + β i<j i=1 ) log x i x j dx spectral measure : λn i = λ N i suitably normalized N µ N = 1 N i=1 δ bλ N i

law of (λ N 1,..., λ N N ) : PN (dx) = 1 Z N (x) β global regime : spectral measure N i=1 dµ(x i ) P N (dx) = 1 Z exp ( N v(x i ) + β i<j i=1 ) log x i x j dx spectral measure : λn i = λ N i suitably normalized µ N = 1 N N i=1 δ bλ N i equilibrium measure

law of (λ N 1,..., λ N N ) : PN (dx) = 1 Z N (x) β global regime : spectral measure N i=1 dµ(x i ) P N (dx) = 1 Z exp ( N v(x i ) + β i<j i=1 ) log x i x j dx spectral measure : λn i = λ N i suitably normalized µ N = 1 N δ bλ N N i i=1 equilibrium measure minimizer of 2 v dν β log x y dν(x)dν(y) weighted logarithmic potential theory

local regime : individual behavior (spacings, extreme values)

local regime : individual behavior (spacings, extreme values) β = 2 orthogonal polynomial method

local regime : individual behavior (spacings, extreme values) β = 2 orthogonal polynomial method P l, l N orthogonal polynomials for µ

local regime : individual behavior (spacings, extreme values) β = 2 orthogonal polynomial method P l, l N orthogonal polynomials for µ N (x) = C n det ( P l 1 (x k ) ) 1 k,l N

local regime : individual behavior (spacings, extreme values) β = 2 orthogonal polynomial method P l, l N orthogonal polynomials for µ N (x) 2 = C 2 n det 2( P l 1 (x k ) ) 1 k,l N

local regime : individual behavior (spacings, extreme values) β = 2 orthogonal polynomial method P l, l N orthogonal polynomials for µ N (x) 2 = C 2 n det 2( P l 1 (x k ) ) 1 k,l N P N (dx) = 1 Z N(x) 2 N i=1 dµ(x i)

local regime : individual behavior (spacings, extreme values) β = 2 orthogonal polynomial method P l, l N orthogonal polynomials for µ N (x) 2 = C 2 n det 2( P l 1 (x k ) ) 1 k,l N P N determinantal random point field structure

local regime : individual behavior (spacings, extreme values) β = 2 orthogonal polynomial method P l, l N orthogonal polynomials for µ N (x) 2 = C 2 n det 2( P l 1 (x k ) ) 1 k,l N P N determinantal random point field structure marginals of P N : determinants of the kernel K N (x, y) = N 1 l=0 P l (x)p l (y)

local regime : individual behavior (spacings, extreme values) β = 2 orthogonal polynomial method P l, l N orthogonal polynomials for µ N (x) 2 = C 2 n det 2( P l 1 (x k ) ) 1 k,l N P N determinantal random point field structure marginals of P N : determinants of the kernel K N (x, y) = N 1 l=0 P l (x)p l (y) orthogonal polynomial ensemble

(common) asymptotics of orthogonal polynomials universal local regime : spacings and edge behavior

(common) asymptotics of orthogonal polynomials universal local regime : spacings and edge behavior top edge behavior largest eigenvalue (particle) fluctuates at the rate (mean) 1/3 limiting Tracy-Widom distribution F TW

(common) asymptotics of orthogonal polynomials universal local regime : spacings and edge behavior top edge behavior largest eigenvalue (particle) fluctuates at the rate (mean) 1/3 limiting Tracy-Widom distribution F TW C. Tracy, H. Widom, P. Deift, X. Zhou, J. Baik, K. Johansson,. Kriecherbauer, K. McLaughlin, P. Miller, S. Venakides, M. Vanlessen, D. Gioev, P. Bleher, A. Its, A. Kuijlaars, M. Shcherbina, L. Pastur...

example : Gaussian Unitary Ensemble (GUE)

example : Gaussian Unitary Ensemble (GUE) dµ(x) = e x2 /2 dx 2π, K N Hermite kernel, λn i = λ N i / N

example : Gaussian Unitary Ensemble (GUE) dµ(x) = e x2 /2 dx 2π, K N Hermite kernel, λn i = λ N i / N spectral measure µ N = 1 N N i=1 δ bλ N 1 4 x i 2π 2 dx on ( 2, +2) Wigner semi-circle law

example : Gaussian Unitary Ensemble (GUE) dµ(x) = e x2 /2 dx 2π, K N Hermite kernel, λn i = λ N i / N spectral measure µ N = 1 N N i=1 δ bλ N 1 4 x i 2π 2 dx on ( 2, +2) Wigner semi-circle law largest eigenvalue λ N N

example : Gaussian Unitary Ensemble (GUE) dµ(x) = e x2 /2 dx 2π, K N Hermite kernel, λn i = λ N i / N spectral measure µ N = 1 N N i=1 δ bλ N 1 4 x i 2π 2 dx on ( 2, +2) Wigner semi-circle law largest eigenvalue λ N N / N 2

example : Gaussian Unitary Ensemble (GUE) dµ(x) = e x2 /2 dx 2π, K N Hermite kernel, λn i = λ N i / N spectral measure µ N = 1 N N i=1 δ bλ N 1 4 x i 2π 2 dx on ( 2, +2) Wigner semi-circle law largest eigenvalue λ N N / N 2 N 1/6[ λ N N 2 N ] F TW Tracy-Widom distribution

example : Gaussian Unitary Ensemble (GUE) dµ(x) = e x2 /2 dx 2π, K N Hermite kernel, λn i = λ N i / N spectral measure µ N = 1 N N i=1 δ bλ N 1 4 x i 2π 2 dx on ( 2, +2) Wigner semi-circle law largest eigenvalue λ N N / N 2 N 1/6[ λ N N 2 N ] F TW Tracy-Widom distribution ( F TW (s) = exp u = 2u 3 + xu s (x s)u(x) 2 dx Painlevé II equation ), s R

example : last passage percolation W N = max π (i,j) π w ij

example : last passage percolation W N = max π (i,j) π w ij µ geometric, K N Meixner kernel, λn i = λ N i /N

example : last passage percolation W N = max π (i,j) π w ij µ geometric, K N Meixner kernel, λn i = λ N i /N spectral measure µ N = 1 N N δ bλ N equilibrium measure on (a, b) i i=1

example : last passage percolation W N = max π (i,j) π w ij µ geometric, K N Meixner kernel, λn i = λ N i /N spectral measure µ N = 1 N N δ bλ N equilibrium measure on (a, b) i i=1 largest particle λ N N W N

example : last passage percolation W N = max π (i,j) π w ij µ geometric, K N Meixner kernel, λn i = λ N i /N spectral measure µ N = 1 N N δ bλ N equilibrium measure on (a, b) i i=1 largest particle λ N N W N/N b

example : last passage percolation W N = max π (i,j) π w ij µ geometric, K N Meixner kernel, λn i = λ N i /N spectral measure µ N = 1 N N δ bλ N equilibrium measure on (a, b) i i=1 largest particle λ N N W N/N b N 1/3[ W N bn ] F TW Tracy-Widom distribution K. Johansson (2000)

Markov operator tools

Markov operator tools for classical orthogonal polynomial ensembles of random matrix and random growth models

Markov operator tools for classical orthogonal polynomial ensembles of random matrix and random growth models non asymptotic results

Markov operator tools for classical orthogonal polynomial ensembles of random matrix and random growth models non asymptotic results spectral description (universal arcsine law)

Markov operator tools for classical orthogonal polynomial ensembles of random matrix and random growth models non asymptotic results spectral description (universal arcsine law) recurrence equations for moments (map enumeration)

Markov operator tools for classical orthogonal polynomial ensembles of random matrix and random growth models non asymptotic results spectral description (universal arcsine law) recurrence equations for moments (map enumeration) recurrence equations for real symmetric models (Gaussian Orthogonal Ensemble)

Markov operator tools for classical orthogonal polynomial ensembles of random matrix and random growth models non asymptotic results spectral description (universal arcsine law) recurrence equations for moments (map enumeration) recurrence equations for real symmetric models (Gaussian Orthogonal Ensemble) non asymptotic tail inequalities for largest eigenvalues (optimal rate)

classical orthogonal polynomial ensembles Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk, Hahn

classical orthogonal polynomial ensembles Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk, Hahn differential equations and operators

classical orthogonal polynomial ensembles Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk, Hahn differential equations and operators mean spectral measure µ N = E ( 1 N i=1 δ λ N ) i

classical orthogonal polynomial ensembles Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk, Hahn differential equations and operators mean spectral measure µ N = E ( 1 N i=1 δ λ N ) i f, µ N = f (x) 1 N K N(x, x) dµ(x) = f N 1 1 Pl 2 N dµ l=0

classical orthogonal polynomial ensembles Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk, Hahn differential equations and operators mean spectral measure µ N = E ( 1 N i=1 δ λ N ) i f, µ N = f (x) 1 N K N(x, x) dµ(x) = f N 1 1 Pl 2 N dµ l=0 example : Gaussian Unitary Ensemble (GUE)

classical orthogonal polynomial ensembles Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk, Hahn differential equations and operators mean spectral measure µ N = E ( 1 N i=1 δ λ N ) i f, µ N = f (x) 1 N K N(x, x) dµ(x) = f N 1 1 Pl 2 N dµ l=0 example : Gaussian Unitary Ensemble (GUE) P l, l N Hermite polynomials for dµ(x) = e x2 /2 dx 2π normalized in L 2 (µ)

f, µ N = f N 1 1 Pl 2 N dµ l=0

f, µ N = f N 1 1 Pl 2 N dµ l=0 preliminary investigation : measure P 2 N dµ

f, µ N = f N 1 1 Pl 2 N dµ l=0 preliminary investigation : measure P 2 N dµ Laplace transform : ϕ(t) = e tx P 2 N dµ, t R

f, µ N = f N 1 1 Pl 2 N dµ l=0 preliminary investigation : measure P 2 N dµ Laplace transform : ϕ(t) = e tx P 2 N dµ, t R Hermite operator : Lf = f x f

f, µ N = f N 1 1 Pl 2 N dµ l=0 preliminary investigation : measure P 2 N dµ Laplace transform : ϕ(t) = e tx P 2 N dµ, t R Hermite operator : Lf = f x f Gaussian integration by parts : f ( Lg)dµ = f g dµ

f, µ N = f N 1 1 Pl 2 N dµ l=0 preliminary investigation : measure P 2 N dµ Laplace transform : ϕ(t) = e tx P 2 N dµ, t R Hermite operator : Lf = f x f Gaussian integration by parts : f ( Lg)dµ = f g dµ LP N = N P N eigenvectors

Laplace transform : ϕ(t) = e tx P 2 N dµ second order differential equation tϕ + ϕ t(t 2 + 4N + 2)ϕ = 0

Laplace transform : ϕ(t) = e tx P 2 N dµ second order differential equation tϕ + ϕ t(t 2 + 4N + 2)ϕ = 0 under the GUE scaling : t t/ N ( λ N i = λ N i / N)

Laplace transform : ϕ(t) = e tx P 2 N dµ second order differential equation tϕ + ϕ t(t 2 + 4N + 2)ϕ = 0 under the GUE scaling : t t/ N ( λ N i = λ N i / N) limiting differential equation t Φ + Φ 4t Φ = 0

Laplace transform : ϕ(t) = e tx P 2 N dµ second order differential equation tϕ + ϕ t(t 2 + 4N + 2)ϕ = 0 under the GUE scaling : t t/ N ( λ N i = λ N i / N) limiting differential equation t Φ + Φ 4t Φ = 0 Φ Laplace transform of the arcsine law dx π 4 x 2 on ( 2, +2)

common behavior, with the limiting arcsine law for the classical orthogonal polynomials of (normalized) measures P 2 N dµ

common behavior, with the limiting arcsine law for the classical orthogonal polynomials of (normalized) measures P 2 N dµ continuous variable : Hermite, Laguerre, Jacobi

common behavior, with the limiting arcsine law for the classical orthogonal polynomials of (normalized) measures P 2 N dµ continuous variable : Hermite, Laguerre, Jacobi discrete variable : Charlier, Meixner, Krawtchouk, Hahn

common behavior, with the limiting arcsine law for the classical orthogonal polynomials of (normalized) measures P 2 N dµ continuous variable : Hermite, Laguerre, Jacobi discrete variable : Charlier, Meixner, Krawtchouk, Hahn varying parameters (in N)

common behavior, with the limiting arcsine law for the classical orthogonal polynomials of (normalized) measures P 2 N dµ continuous variable : Hermite, Laguerre, Jacobi discrete variable : Charlier, Meixner, Krawtchouk, Hahn varying parameters (in N) compact case : A. Maté, P. Nevai, V. Totik (1985)

spectral measure : averaging procedure

spectral measure : averaging procedure GUE E ( f, µ N ) = 1 N N 1 l=0 ( ) l f N x Pl 2 dµ l

spectral measure : averaging procedure GUE E ( f, µ N ) = 1 N N 1 l=0 ( ) l f N x Pl 2 dµ l µ N = 1 N N i=1 δ bλ N i U ξ U uniform, ξ arcsine, independent

spectral measure : averaging procedure GUE E ( f, µ N ) = 1 N N 1 l=0 ( ) l f N x Pl 2 dµ l µ N = 1 N N i=1 δ bλ N i U ξ U uniform, ξ arcsine, independent U ξ semi-circle law on ( 2, +2)

spectral measure : averaging procedure GUE E ( f, µ N ) = 1 N N 1 l=0 ( ) l f N x Pl 2 dµ l µ N = 1 N N i=1 δ bλ N i U ξ U uniform, ξ arcsine, independent U ξ semi-circle law on ( 2, +2) Meixner example (µ geometric) : equilibrium measure

example : last passage percolation W N = max π (i,j) π w ij µ geometric, K N Meixner kernel, λn i = λ N i /N spectral measure µ N = 1 N N δ bλ N equilibrium measure on (a, b) i i=1 largest particle λ N N W N/N b N 1/3[ W N bn ] F TW Tracy-Widom distribution K. Johansson (2000)

spectral measure : averaging procedure GUE E ( f, µ N ) = 1 N N 1 l=0 ( ) l f N x Pl 2 dµ l µ N = 1 N N i=1 δ bλ N i U ξ U uniform, ξ arcsine, independent U ξ semi-circle law on ( 2, +2) Meixner example (µ geometric) : equilibrium measure

spectral measure : averaging procedure GUE E ( f, µ N ) = 1 N N 1 l=0 ( ) l f N x Pl 2 dµ l µ N = 1 N N i=1 δ bλ N i U ξ U uniform, ξ arcsine, independent U ξ semi-circle law on ( 2, +2) Meixner example (µ geometric) : equilibrium measure 1 ( [ ] ) qu(1 + U) ξ + U + q(1 + U) 1 q

moment recursion formula

moment recursion formula E ( e t Tr(X N ) ) = ψ(t) = N 1 e tx l=0 P 2 l dµ, t R differential equation

moment recursion formula E ( e t Tr(X N ) ) = ψ(t) = N 1 e tx l=0 P 2 l dµ, t R differential equation GUE tψ + 3ψ t(t 2 + 4N)ψ = 0 U. Haagerup, S. Thorbjørnsen (2003)

moment recursion formula E ( e t Tr(X N ) ) = ψ(t) = N 1 e tx l=0 P 2 l dµ, t R differential equation GUE tψ + 3ψ t(t 2 + 4N)ψ = 0 U. Haagerup, S. Thorbjørnsen (2003) recurrence equation on moments ( E Tr ( (X N ) p ) ) = N 1 x p l=0 P 2 l dµ, p N

( ap N = E Tr ( (X N ) 2p ) ), X N GUE three term recurrence equation (p + 1)a N p = (4p 2)Na N p 1 + (p 1)(2p 1)(2p 3)aN p 2 J. Harer, D. Zagier (1986)

( ap N = E Tr ( (X N ) 2p ) ), X N GUE three term recurrence equation (p + 1)a N p = (4p 2)Na N p 1 + (p 1)(2p 1)(2p 3)aN p 2 J. Harer, D. Zagier (1986) map enumeration (Wick products)

( ap N = E Tr ( (X N ) 2p ) ), X N GUE three term recurrence equation (p + 1)a N p = (4p 2)Na N p 1 + (p 1)(2p 1)(2p 3)aN p 2 J. Harer, D. Zagier (1986) map enumeration (Wick products) a N p = g 0 ε g (p) N p+1 2g genus series (oriented case)

( ap N = E Tr ( (X N ) 2p ) ), X N GUE three term recurrence equation (p + 1)a N p = (4p 2)Na N p 1 + (p 1)(2p 1)(2p 3)aN p 2 J. Harer, D. Zagier (1986) map enumeration (Wick products) a N p = g 0 ε g (p) N p+1 2g genus series (oriented case) ε 0 (p) Catalan numbers

Markov operator technology similar recursion formulas for the classical orthogonal polynomial ensembles

Markov operator technology similar recursion formulas for the classical orthogonal polynomial ensembles continuous variable : Hermite, Laguerre, Jacobi

Markov operator technology similar recursion formulas for the classical orthogonal polynomial ensembles continuous variable : Hermite, Laguerre, Jacobi discrete variables : Charlier, Meixner, Krawtchouk, Hahn explicit expressions for the (factorial) moments

Gaussian Orthogonal Ensemble β = 1

Gaussian Orthogonal Ensemble β = 1 P N (dx) = 1 Z N (x) N dµ(x i ), x = (x 1,..., x N ) R N i=1 dµ(x) = e x2 /2 dx 2π

Gaussian Orthogonal Ensemble β = 1 P N (dx) = 1 Z N (x) N dµ(x i ), x = (x 1,..., x N ) R N i=1 dµ(x) = e x2 /2 dx 2π Pfaffian instead of determinant

Gaussian Orthogonal Ensemble β = 1 P N (dx) = 1 Z N (x) N dµ(x i ), x = (x 1,..., x N ) R N i=1 dµ(x) = e x2 /2 dx 2π Pfaffian instead of determinant limiting spectral distribution : Wigner semi-circle law Tracy-Widom edge asymptotics

moment recursion formula?

moment recursion formula? mean spectral measure ( E Tr ( f (X N ) )) = f (x) µ N GOE(x)dµ(x)

moment recursion formula? mean spectral measure ( E Tr ( f (X N ) )) = f (x) µ N GOE(x)dµ(x) P l, l N Hermite polynomials µ N GOE = N 1 Pl 2 l=0

moment recursion formula? mean spectral measure ( E Tr ( f (X N ) )) = f (x) µ N GOE(x)dµ(x) P l, l N Hermite polynomials µ N GOE = N 1 l=0 P 2 l + ex2 /4 P N 1 e x 2 /4 P N 1 dµ 1 N odd πn + /4 8 ex2 P N 1 sgn (x y) e x2 /4 (y)p N (y)dµ(y)

Markov operator tools : differential equation on the Laplace transform of the spectral measure

Markov operator tools : differential equation on the Laplace transform of the spectral measure b N p moment recursion formula ( = E Tr ( (X N ) 2p ) ), p N

Markov operator tools : differential equation on the Laplace transform of the spectral measure b N p moment recursion formula ( = E Tr ( (X N ) 2p ) ), p N recursion formula coupled with the moments a N p of the GUE b N p = (4N 2)b N p 1 + 4(2p 2)(2p 3)b N p 2 + a N p (4N 3)a N p 1 (2p 2)(2p 3)a N p 2

effective values a0 N = N a1 N = N 2 GUE a2 N = 2N 3 + N a3 N = 5N 4 + 10N 2 a N 4 = 14N 5 + 70N 3 + 21N b0 N = N b1 N = N 2 + N GOE b2 N = 2N 3 + 5N 2 + 5N b3 N = 5N 4 + 22N 3 + 52N 2 + 41N b4 N = 14N 5 + 93N 4 + 374N 3 + 690N 2 + 509N

b N p ( = E Tr ( (X N ) 2p ) ), X N GOE recursion formula coupled with the moments a N p of the GUE b N p = (4N 2)b N p 1 + 4(2p 2)(2p 3)b N p 2 + a N p (4N 3)a N p 1 (2p 2)(2p 3)a N p 2

b N p ( = E Tr ( (X N ) 2p ) ), X N GOE recursion formula coupled with the moments a N p of the GUE b N p = (4N 2)b N p 1 + 4(2p 2)(2p 3)b N p 2 + a N p (4N 3)a N p 1 (2p 2)(2p 3)a N p 2 b N p five term recurrence equation

b N p ( = E Tr ( (X N ) 2p ) ), X N GOE recursion formula coupled with the moments a N p of the GUE b N p = (4N 2)b N p 1 + 4(2p 2)(2p 3)b N p 2 + a N p (4N 3)a N p 1 (2p 2)(2p 3)a N p 2 b N p five term recurrence equation closed form : I. Goulden, D. Jackson (1997)

b N p ( = E Tr ( (X N ) 2p ) ), X N GOE recursion formula coupled with the moments a N p of the GUE b N p = (4N 2)b N p 1 + 4(2p 2)(2p 3)b N p 2 + a N p (4N 3)a N p 1 (2p 2)(2p 3)a N p 2 b N p five term recurrence equation closed form : I. Goulden, D. Jackson (1997) map enumeration : unoriented case

b N p ( = E Tr ( (X N ) 2p ) ), X N GOE recursion formula coupled with the moments a N p of the GUE b N p = (4N 2)b N p 1 + 4(2p 2)(2p 3)b N p 2 + a N p (4N 3)a N p 1 (2p 2)(2p 3)a N p 2 b N p five term recurrence equation closed form : I. Goulden, D. Jackson (1997) map enumeration : unoriented case duality with Symplectic Ensemble (β = 4)

recursion formulas lead to sharp moment bounds

recursion formulas lead to sharp moment bounds Gaussian Unitary Ensemble (GUE) three term recurrence equation

recursion formulas lead to sharp moment bounds Gaussian Unitary Ensemble (GUE) three term recurrence equation ( E Tr ( (X N ) 2p)) C (4N) p e Cp3 /N 2, p 3 N 2

recursion formulas lead to sharp moment bounds Gaussian Unitary Ensemble (GUE) three term recurrence equation ( E Tr ( (X N ) 2p)) C (4N) p e Cp3 /N 2, p 3 N 2 similar bounds for classical orthogonal polynomial ensembles (β = 2) Gaussian Orthogonal Ensemble (GOE) (β = 1)

non asymptotic small deviation inequalities

non asymptotic small deviation inequalities P ( λ N N 2 N + s N 1/6) F TW (s), s R

non asymptotic small deviation inequalities P ( λ N N 2 N + s N 1/6) F TW (s), s R C 1 e C s3/2 1 F TW (s) C e s3/2 /C (s )

non asymptotic small deviation inequalities P ( λ N N 2 N + s N 1/6) F TW (s), s R C 1 e C s3/2 1 F TW (s) C e s3/2 /C (s ) moment bounds (GUE, GOE) P ( λ N N 2 N + ε ) C e N1/4 ε 3/2 /C, 0 < ε N

non asymptotic small deviation inequalities P ( λ N N 2 N + s N 1/6) F TW (s), s R C 1 e C s3/2 1 F TW (s) C e s3/2 /C (s ) moment bounds (GUE, GOE) P ( λ N N 2 N + ε ) C e N1/4 ε 3/2 /C, 0 < ε N fits the Tracy-Widom asymptotics (ε = s N 1/6 )

non asymptotic small deviation inequalities P ( λ N N 2 N + s N 1/6) F TW (s), s R C 1 e C s3/2 1 F TW (s) C e s3/2 /C (s ) moment bounds (GUE, GOE) P ( λ N N 2 N + ε ) C e N1/4 ε 3/2 /C, 0 < ε N fits the Tracy-Widom asymptotics (ε = s N 1/6 ) deviation inequality on last passage percolation W N K. Johansson (2000)

moment comparison Wigner matrices (independent entries)

moment comparison Wigner matrices (independent entries) example : sign matrix Y N = Y N ij = ±1

moment comparison Wigner matrices (independent entries) example : sign matrix Y N = Y N ij = ±1 asymptotic result : Y. Sinai, A. Soshnikov (1998-99) P ( λ N N (Y ) 2 N + s N 1/6) F TW (s), s R

moment comparison Wigner matrices (independent entries) example : sign matrix Y N = Y N ij = ±1 asymptotic result : Y. Sinai, A. Soshnikov (1998-99) P ( λ N N (Y ) 2 N + s N 1/6) F TW (s), s R ( E Tr ( (Y N ) 2p)) ( E Tr ( (X N ) 2p)), X N GOE

moment comparison Wigner matrices (independent entries) example : sign matrix Y N = Y N ij = ±1 asymptotic result : Y. Sinai, A. Soshnikov (1998-99) P ( λ N N (Y ) 2 N + s N 1/6) F TW (s), s R ( E Tr ( (Y N ) 2p)) ( E Tr ( (X N ) 2p)), X N GOE P ( λ N N (Y ) 2 N + ε ) C e N1/4 ε 3/2 /C, 0 < ε N