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Quantum mixing on large Schreier graphs

Charles Bordenave, Cyril Letrouit, Mostafa Sabri

TL;DR

The paper develops a trace-resolvent framework to establish quantum mixing and quantum ergodicity for sequences of Schreier graphs converging to Cayley graphs with purely absolutely continuous spectrum. By leveraging Benjamini-Schramm/strong distribution convergence and, in stronger forms, the rapid decay property, it delivers QE and QM for both diagonal and certain non-diagonal observables, including matricial extensions to finite coverings. It provides a flexible approach that goes beyond tree-like or lattice settings, with concrete applications to free products, right-angled Coxeter groups, and graph lifts, and it discusses the necessity of its assumptions via detailed counterexamples. The methods connect resolvent estimates, trace identities, and representation theory to deliver quantitative mixing results and extend to non-regular graphs, offering a broad toolkit for spectral delocalization on large graphs.

Abstract

Quantum ergodicity describes the delocalization of most eigenfunctions of Laplace-type operators on graphs or manifolds exhibiting chaotic classical dynamics. Quantum mixing is a stronger notion, additionally controlling correlations between eigenfunctions at different energy levels. In this work, we study families of finite Schreier graphs that converge to an infinite Cayley graph and establish quantum mixing under the assumption that the limiting Cayley graph has absolutely continuous spectrum. The convergence of Schreier graphs is understood in the Benjamini-Schramm sense or in the sense of strong convergence in distribution. Our proofs rely on a new approach to quantum ergodicity, based on trace computations, resolvent approximations and representation theory. We illustrate our assumptions on several examples and provide applications to Schreier graphs associated with free products of groups and right-angled Coxeter groups.

Quantum mixing on large Schreier graphs

TL;DR

The paper develops a trace-resolvent framework to establish quantum mixing and quantum ergodicity for sequences of Schreier graphs converging to Cayley graphs with purely absolutely continuous spectrum. By leveraging Benjamini-Schramm/strong distribution convergence and, in stronger forms, the rapid decay property, it delivers QE and QM for both diagonal and certain non-diagonal observables, including matricial extensions to finite coverings. It provides a flexible approach that goes beyond tree-like or lattice settings, with concrete applications to free products, right-angled Coxeter groups, and graph lifts, and it discusses the necessity of its assumptions via detailed counterexamples. The methods connect resolvent estimates, trace identities, and representation theory to deliver quantitative mixing results and extend to non-regular graphs, offering a broad toolkit for spectral delocalization on large graphs.

Abstract

Quantum ergodicity describes the delocalization of most eigenfunctions of Laplace-type operators on graphs or manifolds exhibiting chaotic classical dynamics. Quantum mixing is a stronger notion, additionally controlling correlations between eigenfunctions at different energy levels. In this work, we study families of finite Schreier graphs that converge to an infinite Cayley graph and establish quantum mixing under the assumption that the limiting Cayley graph has absolutely continuous spectrum. The convergence of Schreier graphs is understood in the Benjamini-Schramm sense or in the sense of strong convergence in distribution. Our proofs rely on a new approach to quantum ergodicity, based on trace computations, resolvent approximations and representation theory. We illustrate our assumptions on several examples and provide applications to Schreier graphs associated with free products of groups and right-angled Coxeter groups.
Paper Structure (59 sections, 39 theorems, 279 equations, 3 figures)

This paper contains 59 sections, 39 theorems, 279 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Setup
  3. Cayley and Schreier graphs.
  4. Quantum ergodicity and quantum mixing on Schreier graphs.
  5. Quantum mixing for asymptotically uncorrelated observables
  6. Quantum mixing under rapid decay property
  7. Quantum mixing under strong convergence
  8. Extension to finite coverings
  9. Applications of the main theorems
  10. Further discussion and open questions
  11. Organization of the paper
  12. Quantum mixing and resolvent estimates
  13. Small scale quantum mixing implies quantum ergodicity and quantum weak mixing
  14. A general bound on quantum variance
  15. Proof ideas
  16. ...and 44 more sections

Key Result

Lemma 1.2

Assume that $\rho_N$ converges in distribution to $\lambda$. For each $N \geq 1$, let $(a_N(x))_{x \in \left[ N \right]} \in \mathbb C^N$ be independent random variables such that $\|a_N\|_{\infty}\leq 1$ almost surely and the expectation of $a_N(x)$ is independent of $x$. Then almost surely, for al

Figures (3)

  • Figure 1: The ball of radius $3$ (in the word length metric) around the origin in the Cayley graph associated to $K_3*K_3$
  • Figure 2: Super-flexible defining diagrams on $|S|=6$, $4$ and $12$ elements. The diagram on the left is a defining diagram for $\mathbb Z_2^{*3} \times \mathbb Z_2^{*3}$, the middle diagram for $\mathbb Z_2 \times \mathbb Z_2^{*3}$.
  • Figure 3: The butterfly graph $G_F$ (left) and part of the tensor product $\mathbb{Z}\times G_F$ (right). A fundamental set is colored in red.

Theorems & Definitions (81)

  • Definition 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8: Relative strong convergence
  • Lemma 2.1: Small scale quantum mixing implies quantum ergodicity and mixing
  • proof
  • ...and 71 more