Universality for random matrices and log-gases László Erdős IST, Austria Ludwig-Maximilians-Universität, Munich, Germany Encounters Between Discrete and Continuous Mathematics Eötvös Loránd University, Budapest, May 22, 2013 With P. Bourgade, A. Knowles, B. Schlein, H.T. Yau, and J. Yin 1
Eugene Wigner (1956): Successive energy levels of large nuclei have universal spacing statistics that can be described by random matrices. [Physics] Freeman Dyson (1962): A gas of particles with a logarithmic interaction reaches local equilibrium very fast. [Statistical Mechanics] De Giorgi-Nash-Moser (1957-60): Uniformly elliptic and parabolic equations in divergence form with rough coefficients have Hölder continuous solutions. [Math] What do these facts have in common? 2
INTRODUCTION Basic question [Wigner]: What can be said about the statistical properties of the eigenvalues of a large random matrix? Do some universal patterns emerge? H = h 11 h 12... h 1N h 21 h 22... h 2N... = (λ 1, λ 2,..., λ N ) eigenvalues? h N1 h N2... h NN N = size of the matrix, will go to infinity. Analogy: Central limit theorem: 1 N (X 1 + X 2 +... + X N ) N(0, σ 2 ) 3
Gaussian Unitary Ensemble (GUE): H = (h jk ) 1 j,k N complex hermitian N N matrix h jk = h kj (for j < k) are complex and h kk are real independent Gaussian random variables with normalization Eh jk = 0, E h jk 2 = 1 N. The eigenvalues λ 1 λ 2... λ N are of order one: (on average) E 1 N i λ 2 i = E 1 N TrH2 = 1 N ij E h ij 2 = 1 Complex hermitian can be replaced with real symmetric or quaternion self-dual (GOE, GSE) 4
Wigner s observations (holds for GUE, GOE, GSE) 1 ρ (x) = 4 x 2 2π i) Density of eigenvalues: Wigner semicircle law 2 2 ii) Level repulsion: Wigner surmise (in the bulk and for GOE) ( P λ i+1 λ i = s ) πs ( N 2 exp π4 ) s2 ds Guessed by a 2x2 matrix calculation Tested against experimental data. 5
Level spacing (gap) histogram for different point processes. NDE Nuclear Data Ensemble, resonance levels of 30 sequences of 27 different nuclei. 6
SINE KERNEL FOR CORRELATION FUNCTIONS Probability density of the eigenvalues: p(x 1, x 2,..., x N ) The k-point correlation function is given by p (k) N (x 1, x 2,..., x k ) := R N k p(x 1,... x k, x k+1,..., x N )dx k+1... dx N Special case: k = 1 (density) ϱ N (x) := p (1) N (x) = R N 1 p(x, x 2,..., x N )dx 2... dx N Used to compute expectation of observables with one eigenvalue: p (k) N E 1 N N i=1 O(λ i ) = O(x)ϱ N (x)dx 1 2π O(x) 4 x 2 dx with higher k computes observables with k evalues. 7
Rescaled correlation functions at energy E p (k) 1 E (x) := [ϱ(e)] kp(k) N ( E + x 1 Nϱ(E), E + x 2 Nϱ(E),..., E + x ) k Nϱ(E) Rescales the gap λ i+1 λ i to O(1). 8
Local level correlation statistics for GUE [Gaudin, Dyson, Mehta] k-point correlation functions are given by k k determinants: lim N p(k) E (x) = det { S(x i x j ) } k i,j=1, sin πx S(x) := πx p (2) ( ) sin π(x1 E (x) 1 x 2 ) 2 (= Level repulsion) π(x 1 x 2 ) The limit is independent of E as long as E < 2 (bulk spectrum) Gap distribution can be obtained from correlation functions by the exclusion-inclusion formula. Wigner surmise is quite precise. Main question: going beyond Gaussian towards universality! There are two almost disjoint directions of generalization: Gaussian is the common intersection. 9
MODEL 1: INVARIANT ENSEMBLES (LOG-GAS) Unitary ensemble: Hermitian matrices with density P(H)dH e Tr V (H) dh Invariant under H UHU 1 for any unitary U Joint density function of the eigenvalues is explicitly known p(λ 1,..., λ N ) = const. (λ i λ j ) β e j V (λ j) i<j classical ensembles β = 1, 2, 4 (orthogonal, unitary, symplectic symmetry classes; GOU, GUE, GSE for Gaussian case, V (x) = x 2 /2) General β > 0: Gibbs measure with inv. temp. β (no matrix): i<j(λ i λ j ) β e βn i V (λ i) e βnh(λ), H = i V (λ i ) 1 N i<j log(λ j λ i ) Prototype of strongly correlated stat.mech. system ( log-gas ). 10
Universality conjecture for log-gases: For any β > 0, the local statistics is independent of V For classical β = 1, 2, 4, correlation fn s can be explicitly computed via large N asymptotics of orthogonal polynomials due to the Vandermonde determinant. Dyson, Gaudin, Mehta: 60-70 s. Gaussian case (Hermite poly) Deift, Pastur-Schcherbina, Bleher-Its, Lubinsky: from 90 s, general case For general β? New method is needed! Main goal: develop new methods in statistical mechanics of strongly correlated systems. 11
MODEL 2: (GENERAL) WIGNER ENSEMBLES H = (h ij ) 1 i,j N, h ji = h ij independent Eh ij = 0, E h ij 2 = s ij, i s ij = 1, c N s ij C N If h ij are i.i.d. then it is called Wigner ensemble. Universality conjecture (Dyson, Wigner, Mehta etc) : If h ij are independent, then the local eigenvalue statistics are the same as for the Gaussian ensembles. Only symmetry type matters. No previous results (apart from Johansson s for hermitian matrices with Gaussian convolution) 12
SOLUTION TO THE UNIVERSALITY CONJECTURES Theorem [E-Schlein-Yau-Yin, 2009-2010] Local ev. statistics is universal for generalized Wigner ensembles in the bulk (and edge). [Tao-Vu, 2010] Hermitian case via moment matching. Theorem [Bourgade-E-Yau, 2011] Let β > 0 and V be real analytic. Then local statistics is universal in the bulk. Formally: weak convergence of the rescaled correlation functions p (k) E (x) universal as N Key ingredient: Uniqueness of the local Gibbs state with log-interaction What else is left? 13
OUTLOOK Subtle points to address (relevant for the edge): 1) Universality at fixed energy (vs. weak limit in E). 2) Gap universality vs. correlation function universality (fixed label vs. fixed energy) 3) Statistics of individual points vs. gap statistics (Gap is stabler!) These issues require new techniques in statistical mechanics Further goals: extend the scope of the models 1) Random Schrödinger, band matrix (depart from mean-field) 2) Random graphs 14
UNIVERSALITY OF GAPS Theorem [E-Yau, 2012] Let β 1 and V real analytic. Then single gap for the log-gas is universal in the bulk. Precise formulation: Let µ V and µ G denote the β-log-gas with V and the Gaussian case. Then Eµ V O ( N(λ i+1 λ i ) ) E µ GO ( N(λ j+1 λ j ) ) CN ε for any fixed label i, j in the bulk. ε is a Hölder regularity exponent from De Giorgi-Nash-Moser theory! (Result also holds for a few consecutive gaps.) Theorem [E-Yau, 2012] The single-gap universality holds for generalized Wigner ensembles in the bulk. Previous result by Tao [2012] for GUE (explicit formulas) + hermitian ensembles with four moments matching GUE. 15
KEY STEPS FOR CORR. FN. UNIV. IN THE WIGNER CASE Step 1. Rigidity estimate on the ev s: Good apriori bound on almost optimal scale up to the edge. Method: System of self-consistent equations for the Green function, control the error by large deviation methods. Step 2. Universality for Wigner matrices with a small ( N ε ) Gaussian component. Method: Modify DBM to speed up its local relaxation, then show that the modification is irrelevant for statistics involving differences of eigenvalues. Step 3. Universality for arbitrary Wigner matrices. Method: Remove the small Gaussian component in Step 2 by resolvent perturbation theory and moment matching. 16
Step 1: RIGIDITY OF EIGENVALUES Let γ k be the classical location, i.e. the k-th quantile of ϱ sc : γk 2 ϱ sc(x)dx = k N Theorem [E-Yau-Yin, 2010] For generalized Wigner ensemble λ k γ k ( means up to (log N) # factors). 1 N 2/3 k 1/3, In the bulk, λ k is rigid on scale 1/N optimal. Deterioration from N 1 to N 2/3 near the bulk is also optimal. Very different from CLT scaling! signature of strong correlations. 17
Step 2: DYSON BROWNIAN MOTION Gaussian convolution matrix interpolates between Wigner and GUE. Evolve the matrix elements with an OU process: dh t = 1 N db t 1 2 H tdt H t e t/2 H 0 + (1 e t ) 1/2 V. dλ i = 1 N db i + ( 1 2 λ i + 1 N j i 1 λ i λ j ) dt Idea: Equilibrium is the invariant ensemble (GUE, etc.) with known local statistics. Global equilibrium is reached in time O(1) (convexity, Bakry-Emery). For local statistics, only local equilibrium needs to be achieved which is much faster. Our main result proves Dyson s conjecture: 18
The picture of the gas coming into equilibrium in two wellseparated stages, with microscopic and macroscopic time scales, is suggested with the help of physical intuition. A rigorous proof that this picture is accurate would require a much deeper mathematical analysis. Freeman Dyson, 1962 on the approach to equilibrium of Dyson Brownian Motion Global equilibrium is reached in time scale of O(1). Local equilibrium was believed to be reached in O(N 1 ). 19
PUTTING TOGETHER We have proved so far universality for Wigner matrices with a single entry distribution having an N ε Gaussian convolution. By resolvent perturbation we can also prove (Step 3) that local stat are stable if four moments (almost) match. To prove the universality for any single entry distribution, we need to approximate it by another one with a small Gaussian components so that the first four moments almost match. This is elementary. 20
Given u, the distribution of matrix element of H, we find a g 0 and a small s so that the small Gaussian convolution of g 0 be close to u 21
OUTLOOK: UNIVERSALITY CONJECTURES Quantum Chaos Conjecture (vague) classical dynamics with potential V chaotic integrable e.v. gap of + V GOE statistics Poisson statistics Geodesic flow: Bohigas-Giannoni-Schmit (1984), Berry-Tabor (1977) Anderson Model (1958): V ω random potential on R d or Z d random Schrödinger operator: H = + λv ω Depending on λ and d, there are two distinct regimes. 22
I: Strong disorder regime: Localization, Poisson local statistics, insulator II: Weak disorder regime: Delocalization, random matrix (GUE, GOE) local statistics, conductor. I. is relatively well understood, II. is not. 23
Conjectured Dichotomy: There are essentially two different behaviors for local eigenvalue statistics of disordered quantum systems: A: Poisson statistics, for systems with little or no correlations. B: Random matrix statistics: for systems with high correlations. Fundamental belief of universality: The macroscopic statistics (like density of states) depend on the models, but the microscopic statistics are independent of the details of the systems except the symmetries. Our results on Wigner matrices verify this conjecture for random matrices, but it is still mean field. Major goal: move towards random Schrödinger (band matrices). 24
SUMMARY 1. We proved bulk universality for β-log-gases with real analytic potential. 2. We proved bulk and edge universality for generalized Wigner matrices (Wigner-Dyson-Mehta conjecture) 3. In both models we proved universality in both senses; averagedenergy and fixed gap. 4. We proved Dyson s conjecture on the dynamics of DBM 5. We established the uniqueness of the Gibbs state for log-gases 25
OPEN QUESTIONS 1. Remove real analyticity from log-gases 2. Prove fixed energy universality beyond Hermitian case 3. Depart from mean field models via band matrices towards random Schrödinger 4. Understand general properties of log-gases, as a universal strongly correlated system. 26