Maximum eigenvalue of symmetric random matrices with dependent heavy tailed entries

TL;DR

This paper studies the maximum eigenvalue of symmetric random matrices with dependent heavy-tailed entries, showing convergence to the Fréchet distribution, extending previous results to cases with tail index less than one.

Contribution

It extends the understanding of eigenvalue behavior in symmetric random matrices with dependent heavy-tailed entries to the case where the tail index is less than one.

Findings

Maximum eigenvalue converges to Fréchet distribution

Spectral radius exhibits similar convergence

Extension of Soshnikov's 2004 result to tail index < 1

Abstract

This paper deals with symmetric random matrices whose upper diagonal entries are obtained from a linear random field with heavy tailed noise. It is shown that the maximum eigenvalue and the spectral radius of such a random matrix with dependent entries converge to the Frech\'et distribution after appropriate scaling. This extends a seminal result of Soshnikov(2004) when the tail index is strictly less than one.