Random matrix ensembles with split limiting behavior

Abstract

We introduce a new family of [Formula: see text] random real symmetric matrix ensembles, the [Formula: see text]-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but [Formula: see text] eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as [Formula: see text]; the remaining [Formula: see text] are tightly constrained near [Formula: see text] and their distribution converges to the [Formula: see text] 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 are able to isolate each regime separately through appropriate choices of weight functions for the eigenvalues and then an analysis of the resulting combinatorics.