Bulk universality of sparse random matrices

TL;DR

This paper proves that the eigenvalue statistics of sparse Erdős-Rényi random graph adjacency matrices match those of the Gaussian Orthogonal Ensemble in the dense regime, demonstrating bulk universality.

Contribution

It establishes bulk universality for sparse Erdős-Rényi matrices in the regime where the average degree grows faster than one, extending to matrices with entries of varying variances.

Findings

Eigenvalue correlation functions match GOE predictions

Eigenvalue gap distributions coincide with GOE

Bulk universality holds for a broad class of sparse matrices

Abstract

We consider the adjacency matrix of the ensemble of Erd\H{o}s-R\'enyi random graphs which consists of graphs on $N$ vertices in which each edge occurs independently with probability $p$. We prove that in the regime $pN \gg 1$ these matrices exhibit bulk universality in the sense that both the averaged $n$-point correlation functions and distribution of a single eigenvalue gap coincide with those of the GOE. Our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances.