Universality of local spectral statistics of random matrices L´ aszl´ o Erd˝ os Ludwig-Maximilians-Universit¨ at, Munich, Germany Abel Symposium, Oslo, Aug 21 2012 With P. Bourgade, A. Knowles, B. Schlein, H.T. Yau, and J. Yin 1
“Perhaps I am now too courageous when I try to guess the distribution of the distances between successive levels (of en- ergies of heavy nuclei). Theoretically, the situation is quite simple if one attacks the problem in a simpleminded fash- ion.The question is simply what are the distances of the characteristic values of a symmetric matrix with random co- e ffi cients.” Eugene Wigner, 1956 Nobel prize 1963 2
INTRODUCTION Basic question [Wigner]: Consider a large matrix whose elements are random variables with a given probability law. What can be said about the statistical properties of the eigenvalues? Do some universal patterns emerge and what determines them? 0 1 h 11 h 12 . . . h 1 N B C h 21 h 22 . . . h 2 N B C H = = ( � 1 , � 2 , . . . , � N ) Eigenvalues? ) B C . . . . . . . . . @ A h N 1 h N 2 . . . h NN N = size of the matrix, will go to infinity. 1 N ( X 1 + X 2 + . . . + X N ) ⇠ N (0 , � 2 ) Analogy: Central limit theorem: p 3
Gaussian Unitary Ensemble (GUE): H = ( h jk ) 1 j,k N hermitian N ⇥ N matrix with p 1 2 h jk = h kj = p ( x jk + iy jk ) and h kk = p x kk N N where x jk , y jk (for j < k ) and x kk are independent standard Gaussian The eigenvalues � 1 � 2 . . . � N are of order one: X X E 1 i = E 1 N Tr H 2 = 1 E | h ij | 2 = 2 � 2 N N i ij at least in average sense. Hermitian can be replaced with symmetric or quaternion self-dual (GOE, GSE) 4
1 (x) = 4 − x 2 ρ 2 π Wigner semicircle law − 2 2 Typical evalue spacing (gap): � i = � i +1 � � i ⇠ 1 N (in the bulk) Observations: i) Semicircle density. ii) Level repulsion. Holds for other symmetry classes GUE, GOE, GSE. For Wishart matrices, i.e. matrices of the form H = AA ⇤ , where the entries of A are i.i.d.: Marchenko-Pastur law 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: ( 238 U ) typical Poisson statistics: Typical random matrix eigenvalues 6
Level spacing (gap) histogram for di ff erent point processes. NDE – Nuclear Data Ensemble, resonance levels of 30 sequences of 27 di ff erent nuclei. 7
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 Z p ( k ) N ( x 1 , x 2 , . . . , x k ) := R N � k p ( x 1 , . . . x k , x k +1 , . . . , x N )d x k +1 . . . d x N Special case: k = 1 (density ) Z % N ( x ) := p (1) N ( x ) = R N � 1 p ( x, x 2 , . . . , x N )d x 2 . . . d x N It allows to compute expectation of observables with one eigenvalue: q Z Z N X E 1 O ( x ) % N ( x )d x ! 1 4 � x 2 d x O ( � i ) = O ( x ) N 2 ⇡ i =1 Higher k computes observables with k evalues. 8
Local level correlation statistics for GUE [Gaudin, Dyson, Mehta] ✓ ◆ n o 2 1 x 1 x 2 [ ⇢ ( E )] 2 p (2) lim E + N ⇢ ( E ) , E + = det S ( x i � x j ) N i,j =1 N ⇢ ( E ) N !1 for any | E | < 2 (bulk spectrum), where S ( x ) := sin ⇡ x ⇡ x ✓ sin ⇡ ( x 1 � x 2 ) ◆ 2 = 1 � (= Level repulsion) ) ⇡ ( x 1 � x 2 ) 9
k -point correlation functions are given by k ⇥ k determinants: ✓ ◆ 1 x 1 x 2 x k [ ⇢ ( E )] k p ( k ) lim E + N ⇢ ( E ) , E + N ⇢ ( E ) , . . . , E + N N ⇢ ( E ) N !1 n o k = det S ( x i � x j ) i,j =1 The limit is independent of E as long as E is in the bulk spectrum, i.e. | E | < 2. Gap distribution (original question of Wigner) is obtained from cor- relation functions by the exclusion-inclusion formula. Main question: going beyond Gaussian towards universality! There are two almost disjoint directions of generalization: Gaussian is the common intersection. 10
GENERALIZATION NO.1: INVARIANT ENSEMBLES Unitary ensemble : Hermitian matrices with density P ( H )d H ⇠ e � Tr V ( H ) d H Invariant under H ! UHU � 1 for any unitary U Joint density function of the eigenvalues is explicitly known ( � i � � j ) � e � P Y j V ( � j ) p ( � 1 , . . . , � N ) = const. i<j classical ensembles � = 1 , 2 , 4 (orthogonal, unitary, symplectic sym- metry classes; GOU, GUE, GSE for Gaussian case, V ( x ) = x 2 / 2) Correlation functions can be explicit computed via orthogonal poly- nomials due to the Vandermonde determinant structure. large N asymptotic of orthogonal polynomials = ) local statistics is indep of V . But density depends on V . 11
GENERALIZATION NO.2: (GENERALIZED) WIGNER ENSEMBLES ¯ H = ( h ij ) 1 i,j N , h ji = h ij independent X E | h ij | 2 = � 2 � 2 E h ij = 0 , ij , ij = 1 , i c ij C N � 2 N p Nh ij | 4+ " < C Moment condition: E | 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 eigenvalues statistics are the same as for the Gaussian ensembles. 12
Several previous results for invariant ensembles Dyson (1962-76), Gaudin-Mehta (1960- ) classical Gaussian ensem- bles via Hermite polynomials General case by Deift etc. (1999), Pastur-Schcherbina (2008), Bleher-Its (1999), Deift etc (2000-, GOE and GSE), Lubinsky (2008) All these results are limited to invariant ensembles and to the clas- sical values of � = 1 , 2 , 4 (OP Method). For non-classical values, there is no underlying matrix ensemble, but the Gibbs measure ( � i � � j ) � e � � N P Y j V ( � j ) p ( � 1 , . . . , � N ) = const. i<j can still be studied (“log-gas”). = ) PROBLEM 1. No previous results for Wigner (apart from Johansson’s for hermitian matrices with Gaussian convolution) Universality of Wigner matrices? = ) PROBLEM 2. 13
PROBLEM 1: NON-CLASSICAL � -ENSEMBLES ( � i � � j ) � e � � N P Y j V ( � j ) p ( � 1 , . . . , � N ) = const. i<j Limit density % is the unique minimizer of Z Z Z I ( ⌫ ) = R V ( t ) ⌫ ( t )d t � R log | t � s | ⌫ ( s ) ⌫ ( t )d t d s. R Theorem [Bourgade-E-Yau, 2011] Let � > 0 and V be real analytic. Let p ( k ) V,N and p ( k ) G,N be the k -point correlation functions for V and for the Gaussian case, V ( x ) = x 2 / 2. Fix E 2 int(supp % ), E 0 2 int(supp % sc ) and " := N � 1 / 2 , then ✓ ◆ Z E + " d x 1 ↵ 1 ↵ k % ( E ) k p ( k ) x + N % ( E ) , . . . , x + V,N 2 " N % ( E ) E � " ✓ ◆ Z E 0 + " d x 1 ↵ 1 ↵ k % sc ( E 0 ) k p ( k ) x + N % sc ( E 0 ) , . . . , x + ! 0 . � G ,N N % sc ( E 0 ) 2 " E 0 � " weakly in ↵ 1 , . . . , ↵ k as N ! 1 . 14
PROBLEM 2: NON-INVARIANT WIGNER ENSEMBLES Theorem [E-Schlein-Yau-Yin, 2009-2010] The bulk universality holds for generalized Wigner ensembles i.e., for | E | < 2, " = N � 1+ � , � > 0 ✓ ◆✓ ◆ Z E + " d x x + b 1 N , . . . , x + b k p ( k ) F,N � p ( k ) lim = 0 weakly µ,N 2 " N N !1 E � " 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. We also have a similar result at the spectral edge (universality of Tracy-Widom distribution) 15
ERD ˝ OS-R´ ENYI RANDOM GRAPHS N = 100 , p = 0 . 01 Adjacency matrix A = ( a ij ), real symmetric with 8 < a ij = � 1 with probability p : q 0 with probability 1 � p , where q := p pN and � = (1 � p ) � 1 / 2 so that Var a ij = N � 1 . Note that E a ij 6 = 0 = ) there is a large eigenvalue. Each column typically has pN = q 2 nonvanishing entries. 16
Theorem [E-Knowles-Yau-Yin, 2011] Bulk and edge universality for enyi sparse matrices with pN � N 2 / 3 . Erd˝ os-R´ Bulk is given by the (analogue of) the sine-kernel. Edge is given by the Airy kernel and Tracy-Widom. Single outlier is Gaussian. 17
RECENT RESULTS ON BULK UNIVERSALITY 1. Hermitian ensemble with C 6 distribution. [EPRSY 2009]. (Brezin-Hikami, contour integral and reverse heat flow approach) 2. Hermitian Wigner ensemble with probability law supported on at least three points [Tao-Vu] (Extension to Bernoulli in [ERSTVY]). Symmetric ensemble with the first 4 moments of matrix elements matching the GOE [Tao-Vu] (4-moment approach) 3. Symmetric ensemble with three point condition [E-Schlein-Yau]. (Dyson Brownian Motion (DBM) flow approach) 4. Generalized symmetric or hermitian Wigner ensembles (the vari- ances were allowed to vary) [E-Yau-Yin]. enyi sparse matrices with pN � N 2 / 3 [E-Knowles-Yau- 5. Erd˝ os-R´ Yin] Similar development for real and complex sample covariance ensem- bles [E-Schlein-Yau-Yin], [Tao-Vu], [Peche], and also for edge univ. 18
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