  We introduce a new family of $N\times N$ 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 $N\to\infty$; the remaining $k$ are tightly constrained near
$N/k$ and their distribution converges to the $k \times k$ 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.
