Sparse block-structured random matrices: universality

Abstract

We study ensembles of sparse block-structured random matrices generated from the adjacency matrix of a Erdös–Renyi random graph with N vertices of average degree Z, inserting a real symmetric d × d random block at each non-vanishing entry. We consider several ensembles of random block matrices with rank r < d and with maximal rank, r = d. The spectral moments of the sparse block-structured random matrix are evaluated for N→∞ , d finite or infinite, and several probability distributions for the blocks (e.g. fixed trace, bounded trace and Gaussian). Because of the concentration of the probability measure in the d→∞ limit, the spectral moments are independent of the probability measure of the blocks (with mild assumptions of isotropy, smoothness and sub-Gaussian tails). The effective medium approximation is the limiting spectral density of the sparse block-structured random ensembles with finite rank. Analogous classes of universality hold for the Laplacian sparse block-structured ensemble. The same limiting distributions are obtained using random regular graphs instead of Erdös–Renyi graphs.