Multifractality and statistical localization in the sparse Barrat-M\'ezard trap model

TL;DR

This paper investigates how the structure of a network influences relaxation modes in the Barrat-Mézard trap model, revealing multifractal, statistically localized wavefunctions that are independent of network topology.

Contribution

It introduces the concept of statistical localization in the model, showing that wavefunction localization is due to spectral properties rather than spatial network structure.

Findings

Wavefunctions are multifractally localized and statistically rather than spatially localized.

Spectral divergences are explained by an effective model.

Power law tails in wavefunctions are characterized and understood.

Abstract

We study within a paradigmatic model for glassy dynamics, the Barrat-M\'ezard trap model, the effect of a nontrivial network structure in the connectivity among traps. Sparseness of this network has recently been shown to lead to divergences in the bulk of the spectrum of the associated master operator [1, 2]. We analyse here specifically the properties of the relaxation modes that contribute to these spectral divergences. We characterize the statistics of the corresponding wavefunctions and demonstrate that they are localized with multifractal properties. The localization patterns are unrelated to the spatial (network) topology, however, and instead fall within the recently introduced class of statistical localization phenomena [3]. To rationalize these results we develop an effective model that successfully explains both the spectral divergences and the power law tails in the wavefunction entries, and provides a clear physical picture of why the localization is statistical rather than spatial.