The Large Connectivity Limit of the Anderson Model on Tree Graphs

TL;DR

This paper rigorously analyzes the Anderson localization transition on infinite regular trees, providing bounds on the critical disorder and confirming predictions in the large connectivity limit.

Contribution

It introduces rigorous bounds on the free energy function related to localization, especially in the large connectivity limit, confirming earlier theoretical predictions.

Findings

Bounds on the critical disorder for localization

Matching upper and lower bounds in the large connectivity limit

Confirmation of early theoretical predictions

Abstract

We consider the Anderson localization problem on the infinite regular tree. Within the localized phase, we derive a rigorous lower bound on the free energy function recently introduced by Aizenman and Warzel. Using a finite volume regularization, we also derive an upper bound on this free energy function. This yields upper and lower bounds on the critical disorder such that all states at a given energy become localized. These bounds are particularly useful in the large connectivity limit where they match, confirming the early predictions of Abou-Chacra, Anderson and Thouless.