Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality of the ESD for a fixed matrix plus small random noise: A stability approach

Philip Matchett Wood

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We study the empirical spectral distribution (ESD) in the limit where $n\to\infty$ of a fixed $n$ by $n$ matrix $M_{n}$ plus small random noise of the form $f(n)X_{n}$, where $X_{n}$ has iid mean 0, variance $1/n$ entries and $f(n)\to0$. It is known for certain $M_{n}$, in the case where $X_{n}$ is iid complex Gaussian, that the limiting distribution of the ESD of $M_{n}+f(n)X_{n}$ can be dramatically different from that for $M_{n}$. We prove a general universality result showing, with some conditions on $M_{n}$ and $f(n)$, that the limiting distribution of the ESD does not depend on the type of distribution used for the random entries of $X_{n}$. We use the universality result to exactly compute the limiting ESD for two families where it was not previously known. The proof of the main result incorporates the Tao–Vu replacement principle and a version of the Lindeberg replacement strategy, along with the newly-defined notion of stability of sets of rows of a matrix.


Nous étudions la distribution spectrale empirique (DSE) dans la limite $n\to\infty$ d’une matrice $M_{n}$, de taille $n$ par $n$, plus un petit bruit aléatoire de la forme $f(n)X_{n}$, où $X_{n}$ a des entrées iid centrées, avec variance $1/n$, et $f(n)\to0$. Il est connu que pour certaines matrices $M_{n}$, dans le cas où $X_{n}$ a des entrées complexes gaussiennes iid, alors la distribution limite de la DSE de $M_{n}+f(n)X_{n}$ peut être radicalement différente de celle de $M_{n}$. Nous prouvons un résultat général d’universalité en montrant, sous quelques conditions sur $M_{n}$ et $f(n)$, que la distribution limite de la DSE ne dépend pas du type de la distribution des entrées aléatoires de $X_{n}$. Nous utilisons ce résultat d’universalité pour calculer exactement la limite de la DSE pour deux familles pour lesquelles le résultat n’était pas connu auparavant. La preuve du résultat principal incorpore le principe de remplacement de Tao–Vu et une version de la stratégie de remplacement de Lindeberg, avec une notion nouvelle de stabilité d’ensembles de lignes d’une matrice.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1877-1896.

Received: 1 April 2014
Revised: 11 June 2015
Accepted: 23 July 2015
First available in Project Euclid: 17 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52

Random matrices Perturbation Universality


Wood, Philip Matchett. Universality of the ESD for a fixed matrix plus small random noise: A stability approach. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1877--1896. doi:10.1214/15-AIHP702.

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