Lattices of matrices (revised)

TL;DR

This paper introduces a lattice-based matrix model with random energies and couplings, analyzing eigenvalue correlations and modeling transitions between localized and extended states within random matrix theory.

Contribution

It presents a new class of lattice matrix models and derives universal eigenvalue correlation behaviors, advancing understanding of localization transitions.

Findings

Eigenvalue correlations exhibit universal behavior.

Model captures transition between localized and extended regimes.

Provides multiple derivations of key results.

Abstract

We study a new class of matrix models, formulated on a lattice. On each site are $N$ states with random energies governed by a Gaussian random matrix Hamiltonian. The states on different sites are coupled randomly. We calculate the density of and correlation between the eigenvalues of the total Hamiltonian in the large $N$ limit. We find that this correlation exhibits the same type of universal behavior we discovered recently. Several derivations of this result are given. This class of random matrices allows us to model the transition between the ''localized" and ''extended" regimes within the limited context of random matrix theory.