RANDOM MATRIX THEORY AND TOEPLITZ DETERMINANTS

Transcription

1 RANDOM MATRIX THEORY AND TOEPLITZ DETERMINANTS David García-García May 13, 2016 Faculdade de Ciências da Universidade de Lisboa

2 OVERVIEW Random Matrix Theory Introduction Matrix ensembles A sample computation: the density of the eigenvalues for GUE Toeplitz determinants Toeplitz operators The Strong Szegő Limit Theorem Minors of Toeplitz matrices

3 RANDOM MATRIX THEORY

4 INTRODUCTION The study of the asymptotics of statistics of the eigenvalues of random matrices

5 INTRODUCTION The study of the asymptotics of statistics of the eigenvalues of random matrices 2d Ising model: correlation functions along lines are determinants of Toeplitz matrices Montgomery s conjecture: the pair correlation between zeros of the Riemann zeta function is the same as the pair correlation between eigenvalues of the Gaussian Unitary Ensemble Baik-Deift-Johansson theorem: the length of the longest increasing subsequence of a random permutation of N numbers converges in distribution to the Tracy-Widom distribution of the largest eigenvalue of a random matrix from the Gaussian Unitary Ensemble Gauge theory, nuclear physics, engineering, telephone encription

6 INTRODUCTION The study of the asymptotics of statistics of the eigenvalues of random matrices 2d Ising model: correlation functions along lines are determinants of Toeplitz matrices Montgomery s conjecture: the pair correlation between zeros of the Riemann zeta function is the same as the pair correlation between eigenvalues of the Gaussian Unitary Ensemble Baik-Deift-Johansson theorem: the length of the longest increasing subsequence of a random permutation of N numbers converges in distribution to the Tracy-Widom distribution of the largest eigenvalue of a random matrix from the Gaussian Unitary Ensemble Gauge theory, nuclear physics, engineering, telephone encription We study matrix ensembles, which are spaces of matrices, often with certain symmetries, with a probability measure defined on the entries of the matrices

7 MATRIX ENSEMBLES The most studied ensembles are the invariant ensembles β = 1 Gaussian Orthogonal Ensemble (GOE), real symmetric matrices β = 2 Gaussian Unitary Ensemble (GUE), complex hermitian matrices β = 4 Gaussian Symplectic Ensemble (GSE), quaternionic self-dual matrices

8 MATRIX ENSEMBLES The most studied ensembles are the invariant ensembles β = 1 Gaussian Orthogonal Ensemble (GOE), real symmetric matrices β = 2 Gaussian Unitary Ensemble (GUE), complex hermitian matrices β = 4 Gaussian Symplectic Ensemble (GSE), quaternionic self-dual matrices together with a probability measure invariant under orthogonal, unitary or symplectic transformations respectively, and such that the entries of the matrices are independent random variables This makes the pdf to be of the form dp(m) = 1 exp( atr(m 2 ) + btr(m) + c)dm (a > 0; b, c R); Z N,β

9 MATRIX ENSEMBLES The most studied ensembles are the invariant ensembles β = 1 Gaussian Orthogonal Ensemble (GOE), real symmetric matrices β = 2 Gaussian Unitary Ensemble (GUE), complex hermitian matrices β = 4 Gaussian Symplectic Ensemble (GSE), quaternionic self-dual matrices together with a probability measure invariant under orthogonal, unitary or symplectic transformations respectively, and such that the entries of the matrices are independent random variables This makes the pdf to be of the form dp(m) = 1 exp( atr(m 2 ) + btr(m) + c)dm (a > 0; b, c R); Z N,β dp(x 1,, x N ) = 1 N N exp a x 2 j + b x j + c x j x i β dx 1 dx N Z N,β j=1 j=1 i<j

10 THE JOINT PROBABILITY FUNCTION: DETERMINANTAL FORM Starting from the Vandermonde determinant i<j (x j x i ) = x 1 x 2 x N x 2 1 x 2 2 x 2 N x N 1 1 x N 1 2 x N 1 N p 0 (x 1 ) p 0 (x 2 ) p 0 (x N ) p 1 (x 1 ) p 1 (x 2 ) p 1 (x N ) p N 1 (x 1 ) p N 1 (x 2 ) p N 1 (x N ),

11 THE JOINT PROBABILITY FUNCTION: DETERMINANTAL FORM Starting from the Vandermonde determinant x 1 x 2 x N p 0 (x 1 ) p 0 (x 2 ) p 0 (x N ) (x j x i ) = x 2 1 x 2 2 x 2 p 1 (x 1 ) p 1 (x 2 ) p 1 (x N ) N i<j, x N 1 1 x N 1 2 x N 1 p N 1 (x 1 ) p N 1 (x 2 ) p N 1 (x N ) N we see that the joint pdf has the determinantal form K N (x 1, x 1) K N (x 1, x 2) K N (x 1, x N ) P(x 1,, x N ) = 1 K N (x 2, x 1 ) K N (x 2, x 2 ) K N (x 2, x N ) det Z N,β, K N (x N, x 1 ) K N (x N, x 2 ) K N (x N, x N ) where N 1 K N (x i, x j ) = e x2 i /2 e x2 j /2 p k (x i )p k (x j ) k=0

12 A SAMPLE COMPUTATION: THE DENSITY OF THE EIGENVALUES FOR GUE If we choose the polynomials p k to satisfy p j (x)p k (x)e x2 dx = c j δ jk, the Hermite polynomials, then K N is a reproducing kernel, and thus the density of the eigenvalues becomes σ N (x) = P(x, x 2,, x N )dx 2 dx N = K N (x, x) = e p 2 k(x) x2 N 1 k=0

13 A SAMPLE COMPUTATION: THE DENSITY OF THE EIGENVALUES FOR GUE If we choose the polynomials p k to satisfy p j (x)p k (x)e x2 dx = c j δ jk, the Hermite polynomials, then K N is a reproducing kernel, and thus the density of the eigenvalues becomes σ N (x) = P(x, x 2,, x N )dx 2 dx N = K N (x, x) = e Thanks to Christoffel-Darboux formula N 1 p k (x)p k (y) = p N 1 p N k=0 p N (x)p N 1 (y) p N 1 (x)p N (y), x y p 2 k(x) x2 N 1 the analysis of this expression is much simplified In the end, this leads to Wigner s semicircle law: σ N (x) 1 π 2N x 2 ( x < 2N) k=0

14 TOEPLITZ DETERMINANTS

15 TOEPLITZ OPERATORS The Fourier coefficients of functions f L 1 (T) are defined as f n = 1 2π π π f(e iθ )e inθ dθ (n Z)

16 TOEPLITZ OPERATORS The Fourier coefficients of functions f L 1 (T) are defined as f n = 1 2π π π f(e iθ )e inθ dθ (n Z) If f L (T), the following expression defines a bounded operator on the Hardy space H 2 (P denotes the projection P : L 2 H 2 ) T f (g) = P(fg) (g H 2 )

17 TOEPLITZ OPERATORS The Fourier coefficients of functions f L 1 (T) are defined as f n = 1 2π π π f(e iθ )e inθ dθ (n Z) If f L (T), the following expression defines a bounded operator on the Hardy space H 2 (P denotes the projection P : L 2 H 2 ) T f (g) = P(fg) (g H 2 ) This is called a Toeplitz operator, and it has the following matrix expression f 0 f 1 f 2 f 1 f 0 f 1 T f = (f i j ) i,j=1 = f 2 f 1 f 0 Its finite truncations T (N) f are called Toeplitz matrices

18 THE STRONG SZEGŐ LIMIT THEOREM Theorem (Szegő, 1952) Let f(e iθ ) = exp( k= c ke ikθ ), with k= k c k 2 < ; then ( ( ) ) lim det T (N) N f /e Nc 0 = exp kc k c k k=1

19 THE STRONG SZEGŐ LIMIT THEOREM Theorem (Szegő, 1952) Let f(e iθ ) = exp( k= c ke ikθ ), with k= k c k 2 < ; then ( ( ) ) lim det T (N) N f /e Nc 0 = exp kc k c k k=1 A consequence: for positive f, the eigenvalues are uniformly distributed 1 ( ) log λ (N) log λ (N) N c 0 = 1 π c k (e ikθ ) dθ N 2π π k=

20 THE STRONG SZEGŐ LIMIT THEOREM Theorem (Szegő, 1952) Let f(e iθ ) = exp( k= c ke ikθ ), with k= k c k 2 < ; then ( ( ) ) lim det T (N) N f /e Nc 0 = exp kc k c k k=1 A consequence: for positive f, the eigenvalues are uniformly distributed 1 ( ) log λ (N) log λ (N) N c 0 = 1 π c k (e ikθ ) dθ N 2π π There are at least 9 proofs of this result! k=

21 MINORS OF TOEPLITZ MATRICES If we consider a minor of a Toeplitz matrix, such as f 0 f 1 f 2 f 3 f 4 f 5 f 1 f 0 f 1 f 2 f 3 f 4 f 2 f 1 f 0 f 1 f 2 f 3 f 3 f 2 f 1 f 0 f 1 f 2 = (f i j ) N i,j=1, f 4 f 3 f 2 f 1 f 0 f 1

22 MINORS OF TOEPLITZ MATRICES If we consider a minor of a Toeplitz matrix, such as f 0 f 1 f 2 f 3 f 4 f 5 f 1 f 0 f 1 f 2 f 3 f 4 f 2 f 1 f 0 f 1 f 2 f 3 f 3 f 2 f 1 f 0 f 1 f 2 = (f i j ) N i,j=1, f 4 f 3 f 2 f 1 f 0 f 1

23 MINORS OF TOEPLITZ MATRICES If we consider a minor of a Toeplitz matrix, such as f 0 f 1 f 2 f 3 f 4 f 5 f 1 f 0 f 1 f 2 f 3 f 4 f 2 f 1 f 0 f 1 f 2 f 3 f 3 f 2 f 1 f 0 f 1 f 2 = (f i j µi ) N i,j=1, f 4 f 3 f 2 f 1 f 0 f 1

24 MINORS OF TOEPLITZ MATRICES If we consider a minor of a Toeplitz matrix, such as f 0 f 1 f 2 f 3 f 4 f 5 f 1 f 0 f 1 f 2 f 3 f 4 f 2 f 1 f 0 f 1 f 2 f 3 f 3 f 2 f 1 f 0 f 1 f 2 = (f i j µi ) N i,j=1, f 4 f 3 f 2 f 1 f 0 f 1 the following formula holds ( ( ) lim det T (N) N f,µ /enc 0 = 1 ) χ µ (π) (σ, π) exp kc k c k, m! π S m where µ is a partition of m, and (σ, π) = where π has γ k cycles of order k (kc k ) γ k, k=1 k=1

25 BIBLIOGRAPHY P Deift and D Gioev Random Matrix Theory: Invariant Ensembles and Universality, Courant Lecture Notes in Mathematics 18 (2009) E Basor Toeplitz determinants, Fisher-Hartwig symbols, and random matrices, in Recent Perspectives in Random Matrix Theory and Number Theory , Cambridge University Press (2005) P Deift, A Its and I Krasovsky Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model Some history and some recent results, Comm Pure Appl Math, 66, (2013) [arxiv: v3 [mathfa]] D Bump and P Diaconis Toeplitz Minors, J Combin Theory Ser A, 97(2), (2002) Beamer template Metropolis, by Matthias Vogelgesang

26 THANK YOU!