Sparse Random Block Matrices : universality

Abstract: We study ensembles of sparse random block matrices generated from the adjacency matrix of a Erd\"os-Renyi random graph with $N$ vertices of average degree $Z$, inserting a real symmetric $d \times d$ random block at each non-vanishing entry. We consider some ensembles of random block matrices with rank $r < d$ and with maximal rank, $r=d$. The spectral moments of the sparse random block matrix are evaluated for $N \to \infty$, $d$ 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 $d \to \infty$ 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 random block ensembles with finite rank. Analogous classes of universality hold for the Laplacian sparse block ensemble. The same limiting distributions are obtained using random regular graphs instead of Erd\"os-Renyi graphs.