Sparse Random Block Matrices : universality

TL;DR

This paper investigates the spectral properties and universality of sparse random block matrices derived from Erd"os-Renyi graphs, demonstrating measure-independent spectral moments and effective medium approximation for large block sizes.

Contribution

It introduces a universality framework for spectral distributions of sparse random block matrices, independent of block distribution details, and extends results to Laplacian and regular graph ensembles.

Findings

Spectral moments become measure-independent as block size increases.

Effective Medium Approximation accurately describes the spectral density.

Universality holds across different graph types and block rank configurations.

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.