Quantum mixing on large Schreier graphs

TL;DR

This paper establishes quantum ergodicity and mixing for sequences of finite Schreier graphs converging to an infinite Cayley graph with absolutely continuous spectrum, using a novel framework that extends to non-regular graphs.

Contribution

It introduces a new approach based on trace identities and representation theory to prove quantum ergodicity for a broad class of Schreier graphs without requiring tree-like structures.

Findings

Quantum ergodicity holds for all orthonormal eigenbases.

Correlations between eigenvectors at different energies vanish asymptotically.

Results apply to various families of Schreier graphs from group actions.

Abstract

We prove quantum ergodicity and quantum mixing for sequences of finite Schreier graphs converging to an infinite Cayley graph whose adjacency operator has absolutely continuous spectrum. Under Benjamini-Schramm convergence (or strong convergence in distribution), we show that correlations between eigenvectors at distinct energies vanish asymptotically when tested against a broad class of local observables. Our results apply to all orthonormal eigenbases and do not require tree-like structure or periodicity of the limiting graph, unlike previous approaches based on non-backtracking operators or Floquet theory. The proof introduces a new framework for quantum ergodicity, based on trace identities, resolvent approximations and representation-theoretic techniques and extends to certain families of non-regular graphs. We illustrate the assumptions and consequences of our theorems on Schreier graphs arising from free products of groups, right-angled Coxeter groups and lifts of a fixed base graph.