Localization properties of the sparse Barrat-M\'ezard trap model

TL;DR

This paper introduces a new method to analyze localization in sparse systems, applied to the Barrat-Mézard trap model, revealing complex localization behaviors and mechanisms distinct from traditional models.

Contribution

The authors develop a cavity theory-based approach to study eigenvector localization in the sparse Barrat-Mézard trap model, highlighting novel localization phenomena.

Findings

Localization varies across the spectrum of relaxation rates.

Localized modes coexist with extended states due to entropic and activation effects.

The route to localization differs from the Anderson model and standard random matrices.

Abstract

Inspired by works on the Anderson model on sparse graphs, we devise a method to analyze the localization properties of sparse systems that may be solved using cavity theory. We apply this method to study the properties of the eigenvectors of the master operator of the sparse Barrat-M\'ezard trap model, with an emphasis on the extended phase. As probes for localization, we consider the inverse participation ratio and the correlation volume, both dependent on the distribution of the diagonal elements of the resolvent. Our results reveal a rich and non-trivial behavior of the estimators across the spectrum of relaxation rates and an interplay between entropic and activation mechanisms of relaxation that give rise to localized modes embedded in the bulk of extended states. We characterize this route to localization and find it to be distinct from the paradigmatic Anderson model or standard random matrix systems.