Localization, anomalous diffusion and slow relaxations: a random distance matrix approach

TL;DR

This paper investigates the spectral properties and localization phenomena of a class of random matrices with elements depending on distances between points, relevant to physical systems like diffusion and glasses, using renormalization and moment methods.

Contribution

It introduces a novel approach combining renormalization group and moment calculations to analyze the spectrum and localization in distance-dependent random matrices across dimensions.

Findings

Eigenvalue distribution density characterized

Eigenmodes exhibit localization properties

Physical implications for diffusion and glasses discussed

Abstract

We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such as the diffusion of particles, slow relaxations in glasses, and scalar phonon localization. Using a combination of a renormalization group procedure and a direct moment calculation, we find the eigenvalue distribution density (i.e., the spectrum) and the localization properties of the eigenmodes, for arbitrary dimension. Finally, we discuss the physical implications of the results.