On spectral properties of large dilute Wigner random matrices O. - PDF document

On spectral properties of large dilute Wigner random matrices O. Khorunzhiy University of Versailles - Saint-Quentin, France We study the spectral norm (maximal eigenvalue λ max ) of n × n random real symmetric matrices H ( n,ρ ) whose elements H ( n,ρ ) , i ≤ j are given by jointly independent ij random variables, similarly to the well-known ensemble of Wigner real symmetric matrices. The difference between H ( n,ρ ) and the Wigner ensemble is that H ( n,ρ ) is equal to 0 with probability 1 − ρ/n ij (dilute version). The concentration parameter ρ = ρ n represents the average number of non-zero elements per row in H ( n,ρ ) . Our results show that in the asymptotic regime when ρ n = n α , n → ∞ , the value α = 2 / 3 is the critical one with respect to the asymptotic behavior of λ max . 1

I.1. Dilute Wigner random matrices = 1 H ( n,ρ ) √ ρ a ij b ( n,ρ ) , 1 ≤ i ≤ j ≤ n, ij ij where { a ij , i ≤ j } are jointly independent r.v. with symmetric probability distribution and  1 , with probability ρ/n b ( n,ρ )   =   ij 0 , with probability 1 − ρ/n     independent r.v. also independent from a ij . i) If ρ = n , then the matrix 1 H ( n ) = √ n a ij ij represents the Wigner ensemble of real symmetric random matrices; ii) 1 ≪ ρ n ≪ n , dilute version of Wigner RM; iii) ρ n = O (1) , n → ∞ , sparse RM. 2

I.2. Semi-circle law (Wigner law) a) Normalized eigenvalue counting function (NCF) σ n ( λ ) = 1  j : λ ( n )   n #  ≤ λ    j    converges as n → ∞ to σ W ( λ ) with the density d 1 � 4 v 2 − λ 2 , dλ σ W ( λ ) = | λ | ≤ 2 v, 2 πv 2 where v 2 = E a 2 ij [E. Wigner, 1955]. b) Spectral norm λ ( n ) max = max k {| λ ( n ) k |} con- verges to 2 v [S. Geman, 1980; Z. F¨ uredi and J. Koml´ os, 1981, V. Girko, 1988; Z.-D. Bai and Y. Q. Yin, 1988]; λ ( n ) max → 2 v as n → ∞ ; in particular,  λ ( n )   P max ≥ 2 v (1 + x )  → 0 , x > 0 .   3

I.3 Dilution of random matrices • Random graphs: symmetric random matrix  1 , with probability ρ/n   B ij =   0 , with probability 1 − ρ/n     is the adjacency matrix of random graph G n ( P n ) with n vertices and with the edge probability P n = ρ/n (P. Ed˝ os and A. R´ enyi, 1959; E. Gilbert, 1959) • Theoretical physics: dilute and sparse disor- dered systems - [Rodgers-Bray, 1988] - [Mirlin-Fyodorov, 1991] • Neural networks theory • etcetera, ... 4

I.4 Semicircle law in dilute RM In H ( n,ρ ) a number of bonds (connections) between cites i and j destroyed, the structure of random matrix is changed. However, if ρ n → ∞ as n → ∞ , the Wigner (or semicircle) law is still valid, σ n,ρ n ( λ ) → σ W ( λ ) with supp( σ ′ W ) = [ − 2 v, 2 v ] - [Rodgers-Bray, 1988] - [K., Khoruzhenko, Pastur, Shcherbina, 1992] - [Cazati-Girko, 1992] - ... What about λ ( n,ρ ) max → ? and  λ ( n,ρ )   P max > 2 v (1 + x n )  ?   5

II. Critical value for the spectral norm Theorem [K., Adv. Probab. 2001] If ρ n = (log n ) 1+ β , β > 0 , then  λ ( n,ρ )   P max > 2 v (1 + x )  → 0 , x > 0 .   If ρ n = (log n ) 1 − β ′ with β ′ > 0 , then n →∞ λ ( n,ρ ) lim sup max = + ∞ . Conclusion: the value ρ ∗ n = log n is critical for the asymptotic behavior of λ ( n,ρ n ) max . Relation with the properties of large random graphs: the edge probability n = log n P ∗ n is the critical one (a sharp threshold) with respect to the connectedness of the random graph G n ( P n ). 6

III.1 Moments of random matrices Since the works of E. Wigner, the moments  2 k , k = 0 , 1 , 2 , . . . 2 k = E 1 M ( n )  H ( n )   n Tr have been used to study the moments of σ n ( λ ), 2 k � λ 2 k dσ n ( λ ) . 2 k = E 1 n M ( n )  λ ( n )   = E �   j  n j =1 In particular, E. Wigner has shown that (2 k )! M ( n ) k !( k + 1)! = v 2 k t k , 2 k → v 2 k where t k are the Catalan numbers. The key idea of S. Geman [ Ann.Probab. , 1980] inspired by U. Grenander is that the limiting behavior of λ ( n ) max can be studied by means of the high moments nM ( n ) 2 k n , n, k n → ∞ . 7

III.2 High moments of Wigner RM 1) k n = O (log n ) [Geman, 1980; Bai-Yin, 1988]  k n t k n , k n = O (log n ) M ( n )  v 2 (1 + ε )   2 k n ≤ implies that  λ ( n )   P max > 2 v (1 + x )  → 0 as n → ∞ ;   2) k n = O ( n 1 / 6 ) [F¨ uredi-Koml´ os, 1981] k n = O ( n 1 / 2 ), k n = o ( n 2 / 3 ) [Ya. G. Sinai and A. Soshnikov, 1998] 3) k n = χn 2 / 3 , χ > 0 [A. Soshnikov, 1999]: nM ( n ) 2 k n → L ( χ ) = L GOE( χ ) , where L ( χ ) does not depend on the details of the probability distribution of a ij ; as a corol- lary, one gets y      λ ( n )   P  max > 2 v  1 +   ≤ G χ ( y ) , y > 0 .       n 2 / 3      The border spectral scale is n − 2 / 3 . 8

IV.1 Dilute Wigner RM Theorem [K., arXiv -2011, in preparation] Let the probability law of a ij has a finite support. Then y      λ ( n,ρ n )   P  > 2 v  1 +   ≤ G χ ( y ) , y > 0     max   n 2 / 3      for ρ n = n 2 / 3(1+ γ ) with any given γ > 0 . Main technical results: A) If ρ n = n 2 / 3(1+ γ ) , γ > 0 , then n →∞ nM ( n,ρ n ) ≤ L ( χ ) , k n = χn 2 / 3 . lim sup 2 k n The upper bound L is universal in the sense that it does not depend on higher moments V 4 , V 6 , . . . , where V 2 l = E | a ij | 2 l , l ≥ 2. B) If ρ n = n 2 / 3 and k n = χn 2 / 3 , then nM ( n,ρ n ) ≥ ℓ ( χ ) (1 + χV 4 ) , n → ∞ . 2 k n 9

IV.2 Critical value for border scale Our results show that the value ρ n = n 2 / 3 represents a critical value for the spectral prop- erties at the border of the spectrum 2 v : - if the dilution is weak , ρ n ≫ n 2 / 3 , then one can expect that the local spectral properties of Dilute RM are the same as for the Wigner RM ensembles; these properties should be in- dependent on the details of the probability dis- tribution of a ij . To prove: correlation function of the moments, Moment version of IPR (K. arXiv, 2010) - if the dilution is moderate , ρ n = O ( n 2 / 3 ), then the asymptotic behavior of λ ( n ) max will de- pend on V 4 = E | a ij | 4 . The same can be true for other local spectral characteristics. - in the case of strong dilution , ρ n ≪ n 2 / 3 , the spectral scale at the border 2 v changes n 2 / 3 to φ ( n ) 1 from ρ , with φ ( n ) = log n (?) 10

V. Relations with the Wigner RM The value of γ in ρ n = n 2 / 3(1+ γ ) depends on the moments V 2 l = E | a ij | 2 l : 3 if V 12+2 φ < ∞ , then γ > ε = 6 + φ. Inversely, if ρ n = n 2 / 3(1+ γ ) , then the universal upper bound of nM ( n,ρ n ) exists provided 2 k n φ > 3 γ − 6 . For the Wigner ensemble, we have ρ n = n , γ = 1 / 2 and then φ > 0, in accordance with the following generalization of earlier results [A. Soshnikov, 1999]; Theorem [K. 2012] If V 12+2 δ exists for any δ > 0 , then for the Wigner RM, n →∞ n M ( n ) k n = χn 2 / 3 , lim 2 k n = L GOE ( χ ) , where L GOE (or L GUE ) does not depend on the moments of V 2 l , l = 2 , ..., 6 and on V 12+2 δ . 11

VI.1 Proof of the upper bound The proof is based on the method of paper [K., Rand. Oper. Stoch. Eqs. 2012], where a modified and improved version of the approach by Ya.G.Sinai and A. Soshnikov completed in [K. and Vengerovsky, arXiv , 2008] is presented. Start point: E. Wigner’s representation of traces   nM 2 k = E  H i 0 ,i 1 · · · H i 2 k − 1 ,i 0   � i 0 ,...,i 2 k − 1  as a sum over 2 k -step trajectories I 2 k = ( i 0 , i 1 , i 2 , . . . , i 2 k − 2 , i 2 k − 1 , i 0 ) . The family {I 2 k } can be separated into the classes of equivalence determined by the num- ber K of self-intersections of the trajectories. When K = 0, the classes are described by the family D 2 k of the Dyck paths: discrete simple walks of 2 k steps in the upper half-plane that start and end at 0. These are equivalent to the rooted half-plane trees. The cardinality |D 2 k | is given by the Catalan number t k . 12

VI.2 Technical questions - Wigner RM, Sinai-Soshnikov approach: the study of simple self-intersections (open ones; V 4 -direct); vertex of maximal exit degree β ; - K., Vengerovsky: proper and imported cells at β ; Brocken-Tree-Structure instants; - K. Rand.Oper.Stoch.Eqs. : V 4 -direct and in- verse edges; generalization to the case of V 2 k - Dilute RM, K. 2012: detailed study of the vertex β of maximal exit degree D ; ¯ D = d 1 + . . . + d L , d L = ( d 1 , . . . , d L ) . ( A ) The following statement improves the tools used by Ya. G. Sinai and A. Soshnikov. D-lemma. Denote by T ( u ) ( ¯ d L ) the fam- k ily of Catalan trees of height u that have L vertices of exit degrees ¯ d L (A). Then √ k k |T ( u ) d L ) | ≤ Le − ηD B ( χ ) t k , u =1 e χu/ ( ¯ � k where η = ln(4 / 3) and B ( χ ) is related with the Brownian bridge. 13