Local law and Tracy-Widom limit for sparse random matrices

TL;DR

This paper establishes local laws and Tracy-Widom fluctuations at the spectral edge for sparse random matrices, including Erdős-Rényi graphs, under certain sparsity conditions, extending universality results.

Contribution

It proves a local law for eigenvalue density near the spectral edge and demonstrates Tracy-Widom fluctuations for extremal eigenvalues in sparse matrices, including Erdős-Rényi graphs.

Findings

Local law for eigenvalue density up to spectral edges

Tracy-Widom fluctuations for extremal eigenvalues under sparsity conditions

Second largest eigenvalue of Erdős-Rényi graph exhibits Tracy-Widom fluctuations for p≫N^{-2/3}

Abstract

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdos-Renyi graph model $G(N,p)$. We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdos-Renyi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue for $p\gg N^{-2/3}$ with a deterministic shift of order $(Np)^{-1}$.