A unifying model for random matrix theory in arbitrary space dimensions

TL;DR

This paper introduces a unifying sparse random block matrix model that captures various random matrix theories across different space dimensions and provides insights into the physical meaning of space dimension in these models.

Contribution

It presents a novel sparse block matrix model that reproduces key random matrix theory models in various limits and clarifies the role of space dimension, with applications to physical systems.

Findings

Eigenvalue spectrum varies with space dimension and coordination number.

Marchenko-Pastur density is recovered as block size goes to infinity.

Model applies to vibrational spectra, wave localization, and disordered systems.

Abstract

A sparse random block matrix model suggested by the Hessian matrix used in the study of elastic vibrational modes of amorphous solids is presented and analyzed. By evaluating some moments, benchmarked against numerics, differences in the eigenvalue spectrum of this model in different limits of space dimension $d$, and for arbitrary values of the lattice coordination number $Z$, are shown and discussed. As a function of these two parameters (and their ratio $Z/d$), the most studied models in random matrix theory (Erdos-Renyi graphs, effective medium, replicas) can be reproduced in the various limits of block dimensionality $d$. Remarkably, the Marchenko-Pastur spectral density (which is recovered by replica calculations for the Laplacian matrix) is reproduced exactly in the limit of infinite size of the blocks, or $d\rightarrow\infty$, which for the first time clarifies the physical meaning of space dimension in these models. The approximate results for $d=3$ provided by our method have many potential applications in the future, from the vibrational spectrum of glasses and elastic networks, to wave-localization, disordered conductors, random resistor networks and random walks.