Mobility Edge for the Anderson Model on Random Regular Graphs

TL;DR

This paper characterizes the phase diagram of the Anderson model on large random regular graphs, showing a finite delocalized spectrum surrounded by localized regions, using spectral analysis and local limit techniques.

Contribution

It proves the existence of a delocalized spectrum interval on random regular graphs with Gaussian disorder, extending Bethe lattice results to finite graphs.

Findings

Finite delocalized spectrum interval identified

Localized regions surround the delocalized spectrum

Spectral properties transferred from Bethe lattice to finite graphs

Abstract

We determine the phase diagram of the Anderson tight-binding model on random regular graphs with Gaussian disorder and sufficiently large degree. In particular, we prove that if the degree is fixed and the number of vertices goes to infinity, the spectrum asymptotically consists of a finite delocalized interval surrounded by two unbounded localized components. Our argument uses a recent description of the spectrum of the tight-binding model on the Bethe lattice (Aggarwal--Lopatto, 2025). By viewing the Bethe lattice as the local limit of a random regular graph, and establishing suitable concentration, eigenvalue-counting, and resolvent estimates, we transfer this characterization of the spectrum of the limiting model to the finite-volume setting.