Spectral statistics of non Hermitian multiparametric Gaussian random matrix ensembles

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. 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'" This motivates us to theoretically analyze the evolution of the ensemble averaged spectral density on the complex plane as well as its local fluctuations with changing system conditions. Our analysis, based on the complexity parameter formulation, reveals the existence of a critical statistics as well as hidden universality in non-ergodic regime of spectral fluctuations. Another important insight given by our analysis is about the similarity of the evolution equation for the spectral angles correlations to that of a circular Brownian ensemble; the detailed existing information about the latter can then be applied to determine those of the former.