The Tracy--Widom law for some sparse random matrices

TL;DR

This paper demonstrates that the largest eigenvalue of a certain class of sparse random matrices, derived from random regular graphs with random sign multiplication, converges to the Tracy--Widom distribution, revealing universal spectral behavior.

Contribution

It establishes the Tracy--Widom law for the largest eigenvalue of sparse random matrices constructed from random regular graphs with random signs, extending universality results.

Findings

Largest eigenvalue converges to Tracy--Widom distribution

Spectral behavior similar to dense random matrices

Universal eigenvalue distribution in sparse regimes

Abstract

Consider the random matrix obtained from the adjacency matrix of a random d-regular graph by multiplying every entry by a random sign. The largest eigenvalue converges, after proper scaling, to the Tracy--Widom distribution.