  We consider a discrete, non-Hermitian random matrix model, which can be
expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We
show that, asymptotically almost surely, the real parts of the eigenvalues of
the non-Hermitian matrix around any fixed index remain interlaced with those of
the anti-symmetric matrix. Along the way, we show that some tools recently
developed to study the eigenvalue distributions of Hermitian matrices extend to
the anti-symmetric setting.
