We analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column/row sum of the matrix elements of its Hermitian operators. The presence of additional constraints besides real-symmetric nature leads to new correlations among their eigenfunctions, hinders a complete delocalization of dynamics and affects the eigenvalues too. The statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of Brownian ensemble appearing between Poisson and Gaussian orthogonal ensemble.

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.

We numerically analyze the random matrix ensembles of real-symmetric matrices with column/row constraints for many system conditions e.g. disorder type, matrix-size and basisconnectivity. The results reveal a rich behavior hidden beneath the spectral statistics and also confirm our analytical predictions, presented in part I of this paper, about the analogy of their spectral fluctuations with those of a critical Brownian ensemble which appears between Poisson and Gaussian orthogonal ensemble.

Contrary to the prevailing notion, we find that the spectrum associated with the extended states in a complex system may belong to the Poisson universality class if the system is subjected to a specific set of constraints. Our results are based on an exact theoretical as well as numerical analysis of column constrained chiral ensembles with circulant off-diagonal blocks and are relevant for a complete understanding of the eigenfunction localization and related physical properties.