Higher order fluctuations of extremal eigenvalues of sparse random matrices

TL;DR

This paper investigates higher order fluctuations of extremal eigenvalues in sparse random matrices, specifically Erdős-Rényi graphs, introducing correction terms to better understand their behavior beyond leading order.

Contribution

It constructs random correction terms to describe sub-leading fluctuations of extremal eigenvalues in sparse matrices, extending previous leading order fluctuation results.

Findings

Established higher order fluctuation corrections for extremal eigenvalues.

Proved a local law up to a shifted spectral edge.

Demonstrated eigenvalue rigidity with these corrections.

Abstract

We consider extremal eigenvalues of sparse random matrices, a class of random matrices including the adjacency matrices of Erd\H{o}s-R\'{e}nyi graphs $\mathcal{G}(N,p)$. Recently, it was shown that the leading order fluctuations of extremal eigenvalues are given by a single random variable associated with the total degree of the graph (Ann. Probab., 48(2):916-962, 2020; Probab. Theory Related Fields, 180:985-1056, 2021). We construct a sequence of random correction terms to capture higher (sub-leading) order fluctuations of extremal eigenvalues in the regime $N^{\epsilon} < pN < N^{1/3-\epsilon}$. Using these random correction terms, we prove a local law up to a shifted edge and recover the rigidity of extremal eigenvalues under some corrections for $pN>N^{\epsilon}$.