Sparse Random Block Matrices

TL;DR

This paper analytically evaluates the spectral moments of large sparse random block matrices structured by Erd"os-Renyi graphs, using free probability techniques to identify spectral densities and laws.

Contribution

It introduces a method to compute spectral moments of sparse block matrices with blocks from classical ensembles, extending analysis to finite and infinite block dimensions.

Findings

Spectral moments are analytically derived for large sparse block matrices.

Identification of probability laws for blocks and parameter limits.

Analytic expressions for spectral density and moments.

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 Erd\"os-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.