Lvy-Khintchine random matrices Paul Jung University of Alabama - PowerPoint PPT Presentation

Lévy-Khintchine random matrices Paul Jung University of Alabama Birmingham September 21, 2014 University of Cincinnati 1/18

Motivation Wigner matrices (’55, ’58). Heavy tailed matrices have i.i.d. entries (up to symmetry) with infinite variance. Cizeau, Bouchaud, Soshnikov, Ben Arous, Guionnet (08); Bordenave, Caputo, Chafai (’11). Adjacency matrices of Erdös-Rényi graphs with p = 1 / n . Rogers, Bray, Zakharevich (’06), Bordenave and Lelarge (’10). General symmetric matrices with symmetric i.i.d. entries: Sum of a row converges weakly as n → ∞ . Limits are infinitely divisible ID ( σ 2 , d , ν ) . 2/18

Annals of Mathematics 1958) 3/18

Normalization for Wigner matrices Empirical (normalized) measure of eigenvalues e j ( ω ) ∈ R : n 1 � δ e j = ESD n . n j = 1 To normalize the entries note that E ( Second Moment ( ESD n )) = E 1 n ) = E 1 n Tr ( A 2 � a ij a ji = n E a 2 ij . n i , j So we need ij ∼ 1 E a 2 n . Instead of normalizing, change the distribution as n varies: E a 2 a ij = a ji ∼ Bernoulli ( λ/ n ) so that ij = λ/ n . 4/18

Normalization for Wigner matrices Empirical (normalized) measure of eigenvalues e j ( ω ) ∈ R : n 1 � δ e j = ESD n . n j = 1 To normalize the entries note that E ( Second Moment ( ESD n )) = E 1 n ) = E 1 n Tr ( A 2 � a ij a ji = n E a 2 ij . n i , j So we need ij ∼ 1 E a 2 n . Instead of normalizing, change the distribution as n varies: E a 2 a ij = a ji ∼ Bernoulli ( λ/ n ) so that ij = λ/ n . 4/18

Main results Suppose each A n has i.i.d. entries up to self-adjointness satisfying: j = 1 A n ( i , j ) d � n = ID ( σ 2 , d , ν ) . lim n →∞ J. (2014) ESD n a.s. weakly converge to a symm. prob. meas. µ ∞ . µ ∞ is the expected spectral measure for vector δ root of a self-adjoint operator on L 2 ( G ) . (Spectral measure for v is defined as d � v , E ( t ) v � ) Wigner matrices: G = N Sparse matrices: G is a Poisson Galton-Watson tree 5/18

Erdős-Rényi random graphs (rooted at 1) ij ∼ λ We need E a 2 n . Adjacency matrices of Erdős-Rényi graphs   0 1 0 0 0   1 0 0 1 0       0 0 0 0 1       0 1 0 0 0       0 0 1 0 0 6/18

Idea of proof (1) As rooted graphs, Erdős-Rényi( λ/ n ) locally converge to a branching process with a Poiss( λ ) offspring distribution. (2) Bordenave-Lelarge (2010) If G n [ 1 ] ⇒ G ∞ [ 1 ] , then one has strong resolvent convergence: for all z ∈ C + , ( zI − A n ) − 1 11 → ( zI − A ∞ ) − 1 11 (3) 11 = E Tr ( zI − A n ) − 1 1 � E ( zI − A n ) − 1 = z − x d E ( ESD n ) n 7/18

ǫ = 1 / 6-close graphs 8/18

Local weak limits of Erdős-Rényi graphs a ij ∼ Bernoulli ( λ/ n ) so the number of offspring is Poisson ( λ ) . Fix k , an offspring in generation bigger than 1, the probability that it’s also a direct offspring (genereation 1) is: P ( 1 ∼ k ) = 1 / n → 0 . Local weak convergence to a Poiss( λ ) branching process 9/18

Weighted-edges case when σ 2 = 0 , d = 0: Aldous’ PWIT 10/18

Free probability: existence under exponential moments By Lévy-It¯ o decomposition, write A n = G n + L n Local weak convergence implies strong resolvent convergence when σ 2 = 0 [handles ( L n ) ]. Voiculescu’s theorem says ( G n ) and ( L n ) are asymptotically free. The LSD of ( A n ) is the free convolution of the LSDs of ( G n ) and ( L n ) . 11/18

What about σ 2 and d ? Interlacing handles drift (rank one perturbation). For the step in the proof where LWC ⇒ Strong resolvent conv. we need n | a 1 j | 2 1 {| a 1 j | 2 ≤ ε } = 0 . ε ց 0 lim lim � n →∞ j = 1 12/18

Problem: edges diverging to infinity 13/18

The Poisson weighted infinite skeleton tree 14/18

Cords to infinity: σ 2 > 0 Distance = resistance on electric networks, and resistance = 1 / conductance The conductance of each parallel edge is “zero”; however, their collective effective conductance is σ and the effective resistance is 1 /σ . Identifying all edges with small conductance to one single point we get that n | a 1 j | 2 1 {| a 1 j | 2 ≤ ε } = 0 . � ε ց 0 lim lim n →∞ j = 1 15/18

Wigner matrices: vacuum state of the free Fock space We can handle infinite second moments in the Gaussian domain of attraction. 16/18

Semicircle Pictures 17/18

Schur complement formula R jj ( z ) d = ( A ∞ − zI ) − 1 Corollary (J. 2014): For z ∈ C + , satisfies 11 − 1   R 00 ( z ) d  z + σ 2 R 11 ( z ) + a 2 � = − j R jj ( z )  j ≥ 2 where { a j } are arrivals of an independent Poisson( ν ) process. 18/18

Thanks for your attention! [AS04] David Aldous and J. Michael Steele. The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on discrete structures , pages 1–72. Springer, 2004. [BAG08] Gérard Ben Arous and Alice Guionnet. The spectrum of heavy tailed random matrices. Communications in Mathematical Physics , 278(3):715–751, 2008. [BCC11a] Charles Bordenave, Pietro Caputo, and Djalil Chafai. Spectrum of large random reversible Markov chains: heavy-tailed weights on the complete graph. The Annals of Probability , 39(4):1544–1590, 2011. [BL10] Charles Bordenave and Marc Lelarge. Resolvent of large random graphs. Random Structures & Algorithms , 37(3):332–352, 2010. [GL09] Adityanand Guntuboyina and Hannes Leeb. Concentration of the spectral measure of large Wishart matrices with dependent entries. Electron. Commun. Probab , 14(334-342):4, 2009. [Zak06] Inna Zakharevich. A generalization of Wigner’s law. Communications in Mathematical Physics , 268(2):403–414, 2006. 19/18