Criticality in Brownian ensembles

A Brownian ensemble appears as a nonequilibrium state of transition from one universality class of random matrix ensembles to another one. The parameter governing the transition is, in general, size-dependent, resulting in a rapid approach of the statistics, in infinite size limit, to one of the two universality classes. Our detailed analysis, however, reveals the appearance of a new scale-invariant spectral statistics, nonstationary along the spectrum, associated with multifractal eigenstates, and different from the two end-points if the transition parameter becomes size-independent. The number of such critical points during transition is governed by a competition between the average perturbation strength and the local spectral density. The results obtained here have applications to wide-ranging complex systems, e.g., those modeled by multiparametric Gaussian ensembles or column constrained ensembles.