Average Spectral Density of Multiparametric Gaussian Ensembles of Complex Matrices

A statistical description of part of a many body system often requires a non-Hermitian random matrix ensemble with nature and strength of randomness sensitive to underlying system conditions. For the ensemble to be a good description of the system, the ensemble parameters must be determined from the system parameters. This in turn makes its necessary to analyze a wide range of multi-parametric ensembles with different kinds of matrix elements distributions. The spectral statistics of such ensembles is not only system-dependent but also non-ergodic as well as non-stationary. A change in system conditions can cause a change in the ensemble parameters resulting an evolution of the ensemble density and it is not sufficient to know the statistics for a given set of system conditions. This motivates us to theoretically analyze a multiparametric evolution of the ensemble averaged spectral density of a multiparametric Gaussian ensemble on the complex plane. Our analysis reveals the existence of an evolutionary route common to the ensembles belonging to same global constraint class and thereby derives a complexity parameter dependent formulation of the spectral density for the non-equilibrium regime of the spectral statistics, away from Ginibre equilibrium limit.