The Limiting Spectral Measure for an Ensemble of Generalized Checkerboard Matrices

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 so that entries in a checkerboard pattern are some fixed constant. In this family, N - k of the eigenvalues are of order square-root of N, and were called bulks, while the rest are tightly contrained around certain multiples of N and were called blips.

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 and with any multiplicity. For example, 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 by k GOE behavior.