Finite Rank Perturbations of Heavy-Tailed Wigner Matrices

TL;DR

This paper investigates the behavior of the largest eigenvalue of heavy-tailed Wigner matrices with finite rank perturbations, revealing universal fluctuations and phase transitions that depend on tail index and vector localization.

Contribution

It extends the analysis of eigenvalue fluctuations to heavy-tailed distributions with infinite fourth moments, uncovering new universality classes and phase transition phenomena.

Findings

Eigenvalue fluctuations are universal for heavy-tailed distributions with index .

A mixed regime at =4 shows dual limiting laws depending on vector localization.

Phase transitions at =1 and within [1, 128/89] for edge cases.

Abstract

One-rank perturbations of Wigner matrices have been closely studied: let $P=\frac{1}{\sqrt{n}}A+\theta vv^T$ with $A=(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric, $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. with centered standard normal distributions, and $\theta>0, v \in \mathbb{S}^{n-1}.$ It is well known $\lambda_1(P),$ the largest eigenvalue of $P,$ has a phase transition at $\theta_0=1:$ when $\theta \leq 1,$ $\lambda_1(P) \xrightarrow[]{a.s.} 2,$ whereas for $\theta> 1,$ $\lambda_1(P) \xrightarrow[]{a.s.} \theta+\theta^{-1}.$ Under more general conditions, the limiting behavior of $\lambda_1(P),$ appropriately normalized, has also been established: it is normal if $||v||_{\infty}=o(1),$ or the convolution of the law of $a_{11}$ and a Gaussian distribution if $v$ is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions $a_{11},$ heavy-tailed with index $\alpha \in (0,4),$ the fluctuations are shown to be universal and dependent on $\theta$ but not on $v,$ whereas a subfamily of the edge case $\alpha=4$ displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether $v$ is localized, each presenting a continuous phase transition at $\theta_0=1, \theta_0 \in [1,\frac{128}{89}],$ respectively. These results build on our previous which analyzes the asymptotic behavior of $\lambda_1(\frac{1}{\sqrt{n}}A)$ in the aforementioned subfamily.