The spectrum of the random environment and localization of noise

TL;DR

This paper studies how the spectral properties of random walks on large, regular graphs with random environments behave, revealing a transition from Gaussian noise to localization as the graph structure changes.

Contribution

It introduces a new analysis of spectral convergence for random walks in random environments on transitive graphs, highlighting a phase transition at degree two.

Findings

Spectral distribution converges to Gaussian noise for large graphs.

Localization occurs at degree two, transitioning from noise to localized eigenstates.

Behavior differs significantly between regular trees and integer lattices.

Abstract

We consider random walk on a mildly random environment on finite transitive d- regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized.