Bulk eigenvalue fluctuations of sparse random matrices

TL;DR

This paper studies the eigenvalue fluctuations of sparse random matrices, including Erdős-Rényi graphs, revealing their joint distributions, CLTs, and correlation structures, which differ from classical Wigner matrices.

Contribution

It identifies the joint limiting distributions and fluctuation behaviors of eigenvalues in sparse matrices, highlighting differences from dense matrix models.

Findings

Eigenvalues satisfy CLTs with normalization N√p

Eigenvalues of same sign are positively correlated, different signs negatively correlated

CLTs established for eigenvalue counting and resolvent trace at mesoscopic scales

Abstract

We consider a class of sparse random matrices, which includes the adjacency matrix of Erd\H{o}s-R\'enyi graphs $\mathcal G(N,p)$ for $p \in [N^{\varepsilon-1},N^{-\varepsilon}]$. We identify the joint limiting distributions of the eigenvalues away from 0 and the spectral edges. Our result indicates that unlike Wigner matrices, the eigenvalues of sparse matrices satisfy central limit theorems with normalization $N\sqrt{p}$. In addition, the eigenvalues fluctuate simultaneously: the correlation of two eigenvalues of the same/different sign is asymptotically 1/-1. We also prove CLTs for the eigenvalue counting function and trace of the resolvent at mesoscopic scales.