Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices.

We investigate the level density σ(x) and the level-spacing distribution p(s) of random matrices M = AF ≠ M{†}, where F is a (diagonal) inner product and A is a random, real, symmetric or complex, Hermitian matrix with independent entries drawn from a probability distribution q(x) with zero mean and finite higher moments. Although not Hermitian, the matrix M is self-adjoint with respect to F and thus has purely real eigenvalues. We find that the level density σ{F}(x) is independent of the underlying distribution q(x) and solely characterized by F, and therefore generalizes the Wigner semicircle distribution σ{W}(x).