Sparse block-structured random matrices: universality

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