Random anti-commuting Hermitian matrices

We consider pairs of anti-commuting [Formula: see text]-by-[Formula: see text] Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of [Formula: see text]-by-[Formula: see text] blocks on which the first matrix has eigenvalues [Formula: see text] and the second has eigenvalues [Formula: see text]. We call [Formula: see text] the skew spectrum of the pair. We derive a formula for the probability density of the skew spectrum, and show that the elements are repelling.