Universal scaling limits at the spectral singularity of structured random matrices

TL;DR

This paper investigates the local spectral behavior of structured Gaussian block matrices at singularities, revealing universal scaling limits influenced by the variance pattern and identifying new singular behaviors at the spectral edge.

Contribution

It characterizes the universal local spectral limits at the spectral singularity for Gaussian block matrices with up to three blocks, linking these limits to the global density and variance pattern.

Findings

Universal scaling limits are determined at spectral singularities for K ≤ 3.

The local spectral density exhibits a logarithmic singularity for K=3.

Scaling limits depend only on the zero pattern of the variance profile.

Abstract

The empirical spectral distribution of Hermitian $K \times K$-block random matrices converges to a deterministic density on the real line with a potential atom at the origin as the dimension of the blocks tends to infinity. In this model the variances of the entries depends on the block and the limiting density is determined by these variances. In the absence of an atom the density is either bounded or has a power law singularity at the origin. We determine all scaling limits of the spectral density on the eigenvalue spacing scale at this singularity for Gaussian matrices with block numbers $K \leq 3$. The appropriate scaling for the universal limit is correctly predicted by the global eigenvalue density. For $K=3$ the local one-point function exhibits an additional logarithmic singularity. The scaling limits depend only on the zero pattern within the variance profile, but not on the values of its positive entries.