Random Matrix Ensembles with Split Limiting Behavior

TL;DR

This paper introduces a new family of random matrix ensembles with a split spectral behavior, where most eigenvalues follow the semi-circle law and a few are constrained near specific values, with explicit limiting distributions.

Contribution

The paper defines the $k$-checkerboard matrix ensembles and derives their explicit limiting spectral measures, revealing a split eigenvalue behavior with novel analytical techniques.

Findings

Bulk eigenvalues follow the semi-circle law.

Remaining eigenvalues are constrained near $N/k$ and follow a hollow GOE distribution.

Results extend to complex and quaternionic matrix ensembles.

Abstract

We introduce a new family of $N\times N$ random real symmetric matrix ensembles, the $k$-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but $k$ eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as $N\to\infty$; the remaining $k$ are tightly constrained near $N/k$ and their distribution converges to the $k \times k$ hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for complex and quaternionic analogues. We isolate the two regimes by using matrix perturbation results and a nonstandard weight function for the eigenvalues, then derive their limiting distributions using a modification of the method of moments and analysis of the resulting combinatorics.