  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.
