On the top eigenvalue of heavy-tailed random matrices

TL;DR

This paper investigates how the largest eigenvalue behaves in heavy-tailed random matrices with power-law entries, revealing different regimes and distributions depending on the tail exponent, including a new limiting distribution at the critical point.

Contribution

It provides a comprehensive analysis of the largest eigenvalue distribution for heavy-tailed matrices, including the derivation of a new limiting distribution at the critical tail exponent mu=4.

Findings

For mu > 4, the largest eigenvalue converges to 2 with Tracy-Widom fluctuations.

For mu < 4, the largest eigenvalue scales as N^{2/mu-1/2} and follows Fréchet statistics.

At mu=4, a new explicit limiting distribution is derived.

Abstract

We study the statistics of the largest eigenvalue lambda_max of N x N random matrices with unit variance, but power-law distributed entries, P(M_{ij})~ |M_{ij}|^{-1-mu}. When mu > 4, lambda_max converges to 2 with Tracy-Widom fluctuations of order N^{-2/3}. When mu < 4, lambda_max is of order N^{2/mu-1/2} and is governed by Fr\'echet statistics. The marginal case mu=4 provides a new class of limiting distribution that we compute explicitely. We extend these results to sample covariance matrices, and show that extreme events may cause the largest eigenvalue to significantly exceed the Marcenko-Pastur edge. Connections with Directed Polymers are briefly discussed.