Quantum ergodicity on large regular graphs

TL;DR

This paper extends the Quantum Ergodicity theorem to large regular graphs, demonstrating that most eigenfunctions are delocalized, using microlocal analysis on trees to mimic manifold proofs.

Contribution

It introduces a quantum ergodicity result for large regular graphs with few short cycles, adapting microlocal analysis techniques from manifold theory.

Findings

Most eigenfunctions are delocalized on large regular graphs

The method applies to expander graphs with few short cycles

Mimics proofs from quantum ergodicity on manifolds

Abstract

We propose a version of the Quantum Ergodicity theorem on large regular graphs of fixed valency. This is a property of delocalization of "most" eigenfunctions. We consider expander graphs with few short cycles (for instance random large regular graphs). Our method mimics the proof of Quantum Ergodicity on manifolds: it uses microlocal analysis on regular trees.