The limiting spectral measure for an ensemble of generalized checkerboard matrices

TL;DR

This paper extends the analysis of generalized checkerboard matrices in random matrix theory, demonstrating how to control and analyze multiple spectral 'blips' at various multiples of N, revealing their limiting spectral distributions.

Contribution

It introduces a method to construct ensembles with customizable blip eigenvalues and develops techniques to analyze their limiting spectral measures, including multiple blips with arbitrary multiplicities.

Findings

Blips can be positioned at any multiples of N with desired multiplicities.

Limiting distributions for single eigenvalue blips converge to a Dirac delta.

Multiple eigenvalue blips exhibit GOE-like behavior.

Abstract

Random matrix theory successfully models many systems, from the energy levels of heavy nuclei to zeros of $L$-functions. While most ensembles studied have continuous spectral distribution, Burkhardt et al introduced the ensemble of $k$-checkerboard matrices, a variation of Wigner matrices with entries in generalized checkerboard patterns fixed and constant. In this family, $N-k$ of the eigenvalues are of size $O(\sqrt{N})$ and were called bulk while the rest are tightly contrained around a multiple of $N$ and were called blip. We extend their work by allowing the fixed entries to take different constant values. We can construct ensembles with blip eigenvalues at any multiples of $N$ we want with any multiplicity (thus we can have the blips occur at sequences such as the primes or the Fibonaccis). The presence of multiple blips creates technical challenges to separate them and to look at only one blip at a time. We overcome this by choosing a suitable weight function which allows us to localize at each blip, and then exploiting cancellation to deal with the resulting combinatorics to determine the average moments of the ensemble; we then apply standard methods from probability to prove that almost surely the limiting distributions of the matrices converge to the average behavior as the matrix size tends to infinity. For blips with just one eigenvalue in the limit we have convergence to a Dirac delta spike, while if there are $k$ eigenvalues in a blip we again obtain hollow $k \times k$ GOE behavior.