Random Matrix: From Wigner to Quantum Chaos Horng-Tzer Yau Harvard University Joint work with P. Bourgade, L. Erd˝ os, B. Schlein and J. Yin 1 Perhaps I am now too courageous when I try to guess the distribution of the distances between successive levels (of energies of heavy nuclei). Theoretically, the situation is quite simple if one attacks the problem in a simpleminded fashion.The question is simply what are the distances of the characteristic values of a symmetric matrix with random coefficients. Eugene Wigner, 1956 Nobel prize 1963 2 Classical Example of Universality—-Central Limit Theorem Gaussian distribution with variance σ 2: x2 exp − µ(x) = √ 2 2σ 2 2πσ Suppose Xj , j = 1, . . . , N are independent random variables with mean zero and variance one. Then the probability distribution of X1 + . . . + XN √ N converges to the Gaussian distribution with variance one. Independence assumption can be weaken, but is crucial. 1 4 Model for independent particles, Poisson statistics: Law of putting K particles independently in an interval of size N so that K/N → ρ is fixed. Question: How to model systems with high correlation? Gaussian unitary ensemble: H = (hjk )1≤j,k≤N hermitian with 2 1 √ √ (xjk + iyjk ) and hjj = xjj hjk = N N and where xjk , yjk , j > k, and xjj are independent centered Gaussian random variables with variance 1/2. Classical ensembles: Gaussian unitary ensemble (GUE), Gaussian orthogonal ensemble (GOE), Gaussian symplectic ensemble(GSE), sample covariance ensembles (Important for statistics applications). 5 • E. Wigner (1955): The excitation spectra of heavy nuclei have the same spacing distribution as the eigenvalues of GOE. Experimental data for excitation spectra of heavy nuclei: typical Poisson statistics: Typical random matrix eigenvalues 6 Global Statistics: Density of state ρ(x) follows the Wigner semicircle law for GUE, GOE and GSE. ρ (x) = −2 1 2π 4 − x2 2 Eigenvalues: λ1 ≤ λ2 ≤ . . . . . . λN , λi+1 − λi ∼ 1/N . Moment method: for all k fixed: 1 1 2 k k TrH → x 4 − x2dx N 2π −2 Z q 7 Wigner surmise (1956): " the gap distribution ∼ πx2 πx exp − 2 4 # dx Idea: guess the probability law from the eigenvalues of 2 × 2 matrices. Correct up to a few percentage points when compared with the GOE gap distribution. 8 9 Probability density of eigenvalues (w.r.t. Lebesgue measure) pN (λ1, ..., λN ) Correlation function for two eigenvalues: Z (2) pN (x1, x2) = pN (x1, x2, ..., xN )dx3...dxN RN −2 Density of states: ρN (x) = Z RN −1 pN (x, x2, ..., xN )dx2...dxN Dyson, Gaudin, Mehta [’60]: local statistics of level correlation a a (2) 1 2 ,E + , lim [ρN (E)]−2pN E + N →∞ N ρN (E) N ρN (E) sin πa (for GUE), |E| < 2 πa Spacing distribution can be computed from the correlation functions. o2 = det S(ai − aj ) , i,j=1 n S(a) = 10 Quantum Chaos conjecture Energy levels in quantum billiards or H = −∆ + V . Laplace equation: −∆ψn = λnψn. Related Question: Random Schrodinger equation V is random by P. Anderson 1958 11 Berry-Tabor conjecture (1977):If the billiard trajectories are integrable, the eigenvalue spacings statistics is given by the Poisson distribution e−xdx. Bohigas-Giannoni-Schmit conjecture (1984):If the billiard is chaotic, the eigenvalue spacings statistics is given by the GOE. 12 Other application of random matrices: tions, biology, finance, traffic Wireless communica- Wishart (1928), Statistics application: Sample covariance ensembles, matrix of the form A+A, A is the data matrix. • Riemann ζ-function: Gap distribution of zeros of ζ function is given by GUE (Montgomery, 1973). Odlyzko data: 13 There are essentially two different behaviors in nature: A: Poisson statistics, for systems with little or no correlations. B: Random matrix statistics: for systems with high correlations. (Edge behavior is different from the bulk.) Fundamental belief of universality of random matrices: The macroscopic statistics depend on the models, but the microscopic statistics are independent of the details of the systems except the symmetries. Classical many-body systems are too hard to solve; Boltzmann’s model of classical statistical physics e−βH . Quantum many-body systems (and highly correlated systems) are too hard to solve; Wigner’s model of random matrices. 14 Unitary ensemble: Hermitian matrices with density P(H)dH ∼ e−βN Tr V (H)dH Invariant under H → U HU −1 for any unitary U (GUE) p(λ1, . . . , λN ) ∼ Y −β (λi − λj )β e P j N V (λj ) ∼ e−N βHN i<j X1 1 X V (λi) − log |λj − λi| HN = N i<j i 2 classical ensembles β = 1, 2, 4 GOE, GUE, GSE. The distribution of the density of λj is given by the equilibrium measure ρV (x)dx which is the minimizer of I(ν) = Z V (t)dν(t) − ZZ log |t − s|dν(s)dν(t) The semicircle law is the case V (x) = x2. Suppose the support of ρV = [A, B]. 15 From VanderMonde determinant structure (β = 2), (2) pN (x1, x2) = C det[KN (xi, xj )]2 i,j=1 √ KN (x, y) = ψ (x)ψN −1(y) − ψN (y)ψN −1(x) . N N x−y ψN orthogonal polynomials w.r.t. e−βV (x)dx. large N asymptotic of orthogonal polynomials =⇒ local eigenvalue statistics indep of V . But density of e.v. depends on V . KN (x + a1, x + a2) → S(a1 − a2), S(a) = sin πa (for GUE), πa Dyson (1962-76), Gaudin-Mehta (1960- ) via Hermite polynomials and general cases by Deift-Its-Zhou (1997), Bleher-Its (1999), Deift-et al (1999-2009) [Riemann-Hilbert method], PasturSchcherbina (1997), Lubinsky (2008) . . . β = 1, 4:(Widom), Deift-Gioev, Kriecherbauer-Shcherbina: Assuming V analytic with some additional assumptions (convex). From 1960 to 2008, all results depend on explicit formulas. Why need different methods for different cases? non-classical β? what about 16 Generalized Wigner Ensembles H = (hkj )1≤k,j≤N , 2, Ehij = 0, E|hij |2 = σij ¯ hji = hij X independent 2 = 1, σ ∼ N −1/2 . σij ij i c 2 ≤ C ≤ σij N N If hij are i.i.d. then it is called Wigner ensembles. (A) Universality conjecture (Wigner-Dyson-Mehta conjecture): If hij are independent, then the local eigenvalues statistics are the same as the Gaussian ensembles. No results up to 2008. More generally, eigenvalue gap distributions depends only on symmetry classes and are independent of models. It is the same for both invariant and non-invariant models (but depend on β which is a symmetry parameter). 18 Theorem [Erd˝ os, Knowles, Schlein, Y, Yin ] The bulk universality holds for generalized Wigner ensembles satisfying (A) and E|xij |4+ε ≤ C, 0 then for −2 < E < 2, b = N −1+ε , ε0 > 0 Z E+b dE 0 (k) b b (k) pF,N − pµ,N E 0 + 1 , . . . , E 0 + k = 0 lim N →∞ E−b 2b N N weakly F µ generalized symmetric matrices GOE generalized hermitian GUE generalized self-dual quaternion GSE real covariance real Gaussian Wishart complex covariance complex Gaussian Wishart Variances can vary in this theorem. Comparison with Tao-Vu later. Generalization to Erd˝ os-Reyni graphs by Knowles-Erd˝ os-Y-Yin (some example of quantum chaos with random data) 19 Theorem[Bourgade-Erdoes-Y 2010-2011] Suppose that V is real analytic and V 00(x) ≥ −C (1) Consider the β-ensemble µβ,V with β > 0. Let E ∈ (A, B) and E 0 ∈ (−2, 2). We have, as N → ∞, α1 1 α2 (2) p x+ ,x + %V (E)2 V,N N %(E) N %(E) α α 1 (2) 1 2 * 0. p x + , x + − G,N 0 2 0 0 %G(E ) N %G(E ) N %G(E ) G stands for the Gaussian with V (x) = x2. 20 Three Steps to the Universality: Non-invariant case Step 1. A priori estimate, Local Semicircle Law. Method: System of self-consistent equations for the Green function, control the error by large deviation methods. Step 2. Universality of Gaussian divisible ensembles √ √ t > 0, H0 is Wigner V is GUE H = 1 − tH0 + tV, i.e., the matrix entries have some Gaussian components. General method based on estimating the convergence to local equilibrium of Dyson Brownian motion. Step 3. Approximation by Gaussian divisible ensembles—– A density argument. Resolvent perturbation expansion to remove the Gaussian part in step 2. 21 Two Steps to the Universality: Invariant case Step 1: A priori estimate (similar to the Step 1 in the noninvariant case) Theorem [Rigidity estimate] For any α > 0 and > 0, there is a constant c > 0 such that for any N ≥ 1 and k ∈ JαN, (1 − α)N K, Pµβ,V |λk − γk | > N −1+ c ≤ ce−cN . (Valid only in the bulk.) Key input: Logarithmic Sobolev inequality, Loop equation. 22 Step 2. Uniqueness of log gas. We order the particles and study the statistics of : {λj : j ∈ I}, I = IL := JL + 1, L + K K, Relabeling: (λ1, λ2, . . . , λN ) := (y1, . . . yL, xL+1, . . . xL+K , yL+K+1, . . . yN ) local equilibrium measure on x with boundary condition y: µV y (dx). µV y (dx) ∼ exp(−N βHy ), X1 1 X Hy (x) = Vy (xi)− log |xj −xi| 2 N i,j∈I i∈I i<j 2 X Vy (x) = V (x) − log |x − yj |. N j6∈I 23 Goal: µy in the bulk is independent of y as N, K → ∞ for ”good boundary conditions”, i.e., Prove the uniqueness of Gibbs measure for good boundary conditions. Tool: Study the relaxation to equilibrium for dynamics with µG y as the invariant measure. This is a generalization of Dyson Brownian Motion. Uniqueness of Gibbs state is established via a dynamical argument, i.e., estimate of relaxation to equilibrium of a ”local DBM”. Fundamental reason of Universality for both invariant and noninvariant: fast relaxation to local equilibrium of DBM. A priori estimate is needed to provide an estimate of local relaxation time. 24 Date 07-08 09May 09May Erdos, Schlein, Y, Yin Local semicircle N −1 scale Prior results scale N −1/2 delocalization of eigenvector Univ. for small N −3/4 Gaussian component v.s. Johansson’s O(1) Gaussian Univ. for Hermitian matrix smooth dist., first univ. 09Jun 09Jun 09Jul Joint paper, hermitian bulk Removes all extra conditions Univ. for symm Wigner case DBM argument appeared Tao and Vu Local semicircle reproved 4 moment thm for e. values Hermitian bulk univ for vanishing 3rd moment. Bernoulli dist. excluded. univ. apart from subexp tail of hij 25 Date 09Nov Erdos, Schlein, Y, Yin Bulk univ for real and complex covariance matrices 09Dec 10Jan 10Mar 10Jul 11Mar 11Apr 11Dec Tao and Vu Bulk univ for complex cov. matrices reproved. Bulk univ for Wigner matrices with varying variances, except Bernoulli Green fn comparison thm Real Bernoulli solved Eigenvalue rigidity proved Dyson conjecture for optimal relaxation proved univ. for sparse matrices finite 4 + moments for Wigner Bulk univ. for general β ensemble with convex V Convexity condition removed Main Open Problems Universality random Schr¨ odinger Universality of band matrices Random band matrices: H is symmetric with independent but not identically distributed entries with mean zero and variance E W |hk`|2 = e−|k−`|/W √ Narrow band, W N =⇒ localization, Poisson statistics √ Broad band, W N =⇒ delocalization, GOE statistics Even the Gaussian case is open. d-regular graphs Prove some examples of quantum chaos. 26

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