# Quantum ergodicity for the Anderson model on regular graphs

## TL;DR

This paper establishes a quantum ergodicity theorem for the Anderson model on regular graphs, showing that eigenfunctions become uniformly distributed in the weak disorder regime, indicating delocalization.

## Contribution

It proves a quantum ergodicity result for the Anderson model on regular graphs, linking spectral delocalization to eigenfunction uniform distribution in the weak disorder regime.

## Key findings

- Eigenfunctions become asymptotically uniformly distributed on regular graphs
- Delocalization occurs in the Anderson model on regular graphs in the weak disorder regime
- Eigenfunctions are delocalized in the Anderson model on the Bethe lattice

## Abstract

We prove a result of delocalization for the Anderson model on the regular tree (Bethe lattice). When the disorder is weak, it is known that large parts of the spectrum are a.s. purely absolutely continuous, and that the dynamical transport is ballistic. In this work, we prove that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed. The precise result is a quantum ergodicity theorem.