Sparse random block matrices

Abstract

The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erdös–Renyi random graph. The blocks are i.i.d. random matrices of the classical ensembles GOE or GUE. The moments are evaluated for finite or infinite dimension of the blocks. The correspondences between sets of closed walks on trees and classes of irreducible partitions studied in free probability together with functional relations are powerful tools for analytic evaluation of the limiting moments. They are helpful to identify probability laws for the blocks and limits of the parameters which allow the evaluation of all the spectral moments and of the spectral density.

Sparse Random Block Matrices : universality

“…4 In the paper [11], where the blocks {X j } are member of the Gaussian Orthogonal Ensemble, we recalled the relevance of factorization to evaluate expectations of multi-matrix products for d finite or infinite. If the entries of the pair of matrices A , B, are stochastically independent…”

“…The limiting spectral function, the Marchenko-Pastur distribution, is obtained if the blocks have any finite rank r ≥ 1, see [8]- [13] for the case r = 1 with random vectors with uniform distribution on the sphere. In Appendix A of [11], it was mentioned how to modify the multiplicities of the products of blocks corresponding to walks on trees, in order to obtain the moments of regular random block matrices. Here too, universality of the expectations implies that the limiting spectral density of the random regular block ensemble is not dependent on the probability distribution of the block entries.…”

“…In recent years, an ensemble of sparse random block matrices was considered, where the entry A i,j of the random matrix is a real symmetric d × d random matrix X i,j , [7], [8], [9], [10], [11],…”

“…The interaction between any pair of nodes belonging to the same community is different from the interaction of any pair of nodes belonging to different communities. We recalled in Appendix B of [11] some similarities of the stochastic block model and the Equitable random graph with the sparse block matrix ensembles here studied. References quoted there may introduce readers to the vast literature of complex systems modeled on random networks.…”

Sparse block-structured random matrices: universality

“…More pertinent to this paper, the moments of the spectral density of the limiting (N → ∞) adjacency matrix A N in equation (1.1) were carefully studied and recursion relations for them were obtained [5,6]. Remarkably, the knowledge of all the spectral moments, at least in principle, was not sufficient to obtain the spectral density, possibly because, as it is indicated in [11], the moments correspond to a class of walks on random trees that was not enumerated.…”