Random matrix ensembles with split limiting behavior

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 are able to isolate each regime separately through appropriate choices of weight functions for the eigenvalues and then an analysis of the resulting combinatorics.