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

We investigate the level-density $\sigma (x)$ and level-spacing distribution $p(s)$ of random matrices $M=AF\mathrm{\ne }{M}^{†}$ 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 ${\sigma }_F(x)$ is independent of the underlying distribution $q(x)$, is solely characterized by $F$, and therefore generalizes Wigner’s semicircle distribution ${\sigma }_W(x)$. We find that the level-spacing distributions $p(s)$ are independent of $q(x)$, are dependent upon the inner-product $F$ and whether $A$ is real or complex, and therefore generalize the Wigner’s surmise for level spacing. Our results suggest $F$-dependent generalizations of the well-known Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE) classes.