Return probability for the Anderson model on the random regular graph

TL;DR

This paper investigates the return probability in the Anderson model on random regular graphs, providing evidence for two distinct phases—ergodic and nonergodic—by analyzing decay behaviors and benchmarking with random matrix models.

Contribution

It introduces a method to distinguish ergodic and nonergodic phases via return probability analysis, supported by benchmarking and application to the Anderson model.

Findings

Identifies polynomial decay in ergodic phase

Finds stretched exponential decay in nonergodic phase

Provides evidence for two phases in the Anderson model

Abstract

We study the return probability for the Anderson model on the random regular graph and give evidence of the existence of two distinct phases: a fully ergodic and nonergodic one. In the ergodic phase, the return probability decays polynomially with time with oscillations, being the attribute of the Wigner-Dyson-like behavior, while in the nonergodic phase the decay follows a stretched exponential decay.We give a phenomenological interpretation of the stretched exponential decay in terms of a classical random walker. Furthermore, comparing typical and mean values of the return probability, we show how to differentiate an ergodic phase from a nonergodic one. We benchmark this method first in two random matrix models, the power-law random banded matrices, and the Rosenzweig-Porter matrices, which host both phases. Second, we apply this method to the Anderson model on the random regular graph to give further evidence of the existence of the two phases.