Quantum mixing on large Schreier graphs

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.

💡 Research Summary

The paper establishes quantum ergodicity and quantum mixing for large families of finite Schreier graphs that converge, in the Benjamini‑Schramm sense, to an infinite Cayley graph whose adjacency operator has purely absolutely continuous spectrum on a fixed interval. The authors work with a finitely generated group Γ and a symmetric generating set S, forming the Cayley graph Cay(Γ,S). For each N they consider a permutation representation ρ_N: Γ → S_N, which yields the Schreier graph Sch(Γ,S,ρ_N) on the vertex set