Quantum ergodicity on large graphs

TL;DR

This paper presents three concise and diverse proofs of quantum ergodicity for eigenfunctions of the Laplacian on large regular graphs, aiming to adapt these methods to other models in mathematical physics.

Contribution

It introduces three novel, shorter proofs of quantum ergodicity on large regular graphs, each providing unique insights and potential for adaptation to related models.

Findings

Three different proofs of quantum ergodicity are established.

The proofs are shorter and more diverse than the original.

Extension to anisotropic random walks is achieved.

Abstract

We give three different proofs of the main result of Anantharaman-Le Masson, establishing quantum ergodicity -- a form of delocalization --for eigenfunctions of the laplacian on large regular graphs of fixed degree. These three proofs are much shorter than the original one, quite different from one another, and we feelthat each of the four proofs sheds a different light on the problem. The goal of this exploration is to find a proof that could be adapted for othermodels of interest in mathematical physics, such as the Anderson model on large regular graphs, regular graphs with weighted edges, or possibly certain models of non-regular graphs.A source of optimism in this direction is that we are able to extend the last proof to the case of anisotropic random walks on large regular graphs.