Random matrix ensembles with split limiting behavior

Abstract

We introduce a new family of N × 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 → ∞; the remaining k are tightly constrained near N/k and their distribution converges to the k × 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.

The Limiting Spectral Measure for an Ensemble of Generalized Checkerboard Matrices

“…The proof is by standard combinatorial arguments similar to the one in [2]. Weyl's Inequality [3] implies that if the spectral radius of P is O(f ) then the size of the perturbations are O(f ) as well.…”

“…What makes the checkerboard ensemble in [2] interesting is that the eigenvalues of a matrix from the ensemble almost surely fall into two separate regimes. With our generalization we can exploit the freedom to choose different constants to force the eigenvalues to fall into more regimes.…”

“…As in [2], we refer to the N − x eigenvalues that are on the order of √ N as the eigenvalues in the bulk, while for each distinct w i , the k i eigenvalues near N w i /k are called the eigenvalues in the blips. We study the eigenvalue distribution of each regime.…”

“…We generalize [2] by allowing the constant w to take different values. While the Checkerboard ensembles in [2] only allow one blip for each ensemble, the generalized Checkerboard ensembles allow arbitrarily many blips for each ensemble. Moreover, we have control over the positions of these blips.…”

“…This paper is a sequel to [2], where they introduce ensembles of checkerboard matrices which also have a nice, closed-form expression for their limiting distribution. The spectrum splits into two; N − k of the eigenvalues are of size √ N (called the bulk eigenvalue) and converges to a semicircle, while k of the eigenvalues (called the blip eigenvalues) are of order N and converge to the spectral distribution of a k ×k hollow Gaussian orthogonal ensemble.…”