Local law and Tracy-Widom limit for sparse sample covariance matrices

TL;DR

This paper establishes a local spectral law and Tracy-Widom fluctuations for the extremal eigenvalues of sparse sample covariance matrices, including bipartite Erdős-Rényi graphs, under certain sparsity conditions.

Contribution

It proves a local law for eigenvalue density and Tracy-Widom limit for extremal eigenvalues in sparse covariance matrices, with explicit spectral edge shifts.

Findings

Eigenvalue density follows a local law up to the spectral edge.

Extremal eigenvalues follow Tracy-Widom distribution under sparsity conditions.

Second largest eigenvalue exhibits Tracy-Widom fluctuations when connection probability exceeds a threshold.

Abstract

We consider spectral properties of sparse sample covariance matrices, which includes biadjacency matrices of the bipartite Erd\H{o}s-R\'enyi graph model. We prove a local law for the eigenvalue density up to the upper spectral edge. Under a suitable condition on the sparsity, we also prove that the limiting distribution of the rescaled, shifted extremal eigenvalues is given by the GOE Tracy-Widom law with an explicit formula on the deterministic shift of the spectral edge. For the biadjacency matrix of an Erd\H{o}s-R\'enyi graph with two vertex sets of comparable sizes $M$ and $N$, this establishes Tracy-Widom fluctuations of the second largest eigenvalue when the connection probability $p$ is much larger than $N^{-2/3}$ with a deterministic shift of order $(Np)^{-1}$.