Lévy-Khintchine Random Matrices Paul Jung University of Alabama Birmingham Western States Mathematical Physics Meeting February 15, 2016 1/18
α -Stable limit laws (heavy tails) Finite variance is important in the CLT: 1 α ) ∼ 1 If P ( | X i | > Cn n for 0 < α < 2 then for all x X 1 + X 2 + · · · + X n �� � � � √ n � P � < x → 0 � � � On the otherhand, � X 1 + X 2 + · · · + X n � P < x → F ( x ) n 1 /α where F ( x ) is an α -Stable cdf. When α = 1, F ( x ) is the Cauchy distribution function. 2/18
α -Stable limit laws (heavy tails) Finite variance is important in the CLT: � 1 1 1 � ∼ 1 α > Cn α / n If P | X i | / n n for 0 < α < 2 then for all x α X 1 + X 2 + · · · + X n �� � � � √ n � P � < x → 0 � � � On the otherhand, � X 1 + X 2 + · · · + X n � P < x → F ( x ) n 1 /α where F ( x ) is an α -Stable cdf. When α = 1, F ( x ) is the Cauchy distribution function. 2/18
3 Different normalizations CLT: n 1 E X 2 i = σ 2 � √ nX i ∼ N ( 0 , σ 2 ) . � i = 1 Poisson: � n P ( X i = 1 ) = 1 / n � X i ∼ Poisson ( λ ) . � P ( X i = 0 ) = 1 − 1 / n i = 1 α -Stable: n α ) ∼ 1 1 1 � P ( | X i | > Cn n 1 /α X i ∼ Stable α ( σ ) . � n i = 1 3/18
Wigner’s theorem (1955, 58) Matrix A n has i.i.d. entries up to symmetry A n ( i , j ) = A n ( j , i ) and such that E A n ( i , j ) 2 < ∞ Normalized empirical measure of eigenvalues e j ∈ R converges n 1 1 � 4 − x 2 dx � δ e j = ESD n = ⇒ 2 π n j = 1 4/18
Normalization for Wigner matrices n 1 1 � 4 − x 2 dx � δ e j = ESD n = ⇒ 2 π n j = 1 What is the correct scaling? Tightness of probability measures: Second Moment ( ESD n ) = 1 n ) = 1 A n ( i , j ) 2 ≃ nA n ( i , j ) 2 . n Tr ( A 2 � n i , j So we need � 1 � A n ( i , j ) 2 = O . n 5/18
Normalization for Wigner matrices n 1 1 � 4 − x 2 dx � δ e j = ESD n = ⇒ 2 π n j = 1 What is the correct scaling? Tightness (in expectation) of random probability measures: E ( Second Moment ( ESD n )) = E 1 n Tr ( A 2 n ) = n E A n ( i , j ) 2 . So we need � 1 � E A n ( i , j ) 2 = O . n Instead of normalizing, change the distribution as n varies: E A n ( i , j ) 2 = λ/ n . A n ( i , j ) ∼ Bernoulli ( λ/ n ) so that 5/18
3 Different normalizations for random matrices Wigner: E A n ( i , j ) 2 ∼ 1 1 √ nN ( 0 , σ 2 ) . A n ( i , j ) ∼ � n Sparse: E A n ( i , j ) 2 ∼ 1 � λ � A n ( i , j ) ∼ Poisson . � n n Heavy-tailed: α ) ∼ 1 1 1 P ( | A n ( i , j ) | > Cn A n ( i , j ) ∼ n 1 /α Stable α ( σ ) . � n 6/18
Infinite divisibility X (symmetric) is infinitely divisible if for every n X d = A n ( 1 , 1 ) + A n ( 1 , 2 ) + · · · + A n ( 1 , n ) ( i . i . d . ) and it is determined by ( σ 2 , ν ) satisfying, when X d = − X , � − 1 � � E e i θ X = exp 2 σ 2 θ 2 + ( e i θ x − 1 ) ν ( dx ) R 7/18
Existence of the LSD Suppose each A n has i.i.d. entries up to self-adjointness such that for each i : j = 1 A n ( i , j ) d � n = ID ( σ 2 , ν ) . lim n →∞ J. (2015) With probability 1, ESD n weakly converges 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 ) . Wigner matrices: G = N Sparse matrices: G is a Poisson Galton-Watson tree 8/18
ESD of heavy-tailed and gamma random matrices Histogram of Cauchy Histogram of Gamma 350 250 300 200 250 200 150 Frequency Frequency 150 100 100 50 50 0 0 -40 -20 0 20 40 -4 -2 0 2 4 Cauchy Gamma 9/18
Proof sketch: Existence of the LSD (1) As rooted graphs, Erdős-Rényi( λ/ n ) locally converge to a branching process with Poisson( λ ) offspring distribution. (2) Bordenave-Lelarge (2010) If G n [ 1 ] ⇒ G ∞ [ 1 ] , then for all z ∈ C + , ( A n − zI ) − 1 11 → ( A ∞ − zI ) − 1 11 := � δ 1 , ( A ∞ − zI ) − 1 δ 1 � , where A ∞ is an adjacency operator on L 2 ( G ∞ ) . (3) 11 = E 1 1 � n Tr ( A n − zI ) − 1 = E ( A n − zI ) − 1 x − z d E ( ESD n ) . 10/18
Local weak limits of Erdős-Rényi graphs Fixing an offspring k with depth 2 or more: P ( 1 is connected to k ) = λ/ n → 0 . Local weak convergence to a Poiss( λ ) branching process 11/18
Topology for rooted weighted graphs (locally-finite) Such graphs form a Polish space so weak convergence makes sense. 12/18
Aldous’ Poisson weighted infinite tree For heavy-tailed matrices, the weights 1 / | A n ( i , j ) | are arrivals of a Poisson process of intensity | x | α − 1 dx . 13/18
The Poisson weighted infinite skeleton tree 14/18
The cords to infinity: σ 2 > 0 Interpret the weights as the lengths of edges. Thus, v and ∞ v are infinitely far apart, but have infinitely many parallel edges between. Conductance of each parallel edge is zero, but their collective effective conductance is σ . 15/18
Corollary We handle infinite variance in the Gaussian domain of attraction. 16/18
Schur complement formula for the limiting operators Corollary (J. 2014) R jj ( z ) d = ( A ∞ − zI ) − 1 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. Proof follows from the resolvent identity. 17/18
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. [CB94] P. Cizeau and J.P. Bouchaud. Theory of Lévy matrices. Physical Review E , 50(3):1810, 1994. [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. 18/18
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