Quantum Mixing and Benjamini-Schramm Convergence of Hyperbolic Surfaces

TL;DR

This paper extends quantum mixing results to large-scale hyperbolic surfaces, using hyperbolic wave equations and exponential mixing, applicable to arithmetic and random surfaces of large genus.

Contribution

It introduces a novel approach based on hyperbolic wave equations and exponential mixing, avoiding traditional averaging operators, to analyze quantum mixing on large hyperbolic surfaces.

Findings

Establishes a large-scale quantum mixing theorem for hyperbolic surfaces.

Applicable to both arithmetic and Weil--Petersson random surfaces.

Provides a more complete understanding of asymptotic observable behavior.

Abstract

We study compact hyperbolic surfaces and multiplication observables, establishing a large-scale analogue of Zelditch's quantum mixing theorem with hypotheses that hold for both arithmetic and Weil--Petersson random surfaces of large genus. This complements the large-scale quantum ergodicity theorems of Le Masson and Sahlsten, which themselves are large-scale analogues of the quantum ergodicity theorem of Shnirelman, Zelditch, and Colin de Verdi\`{e}re, thereby providing a more complete picture of the asymptotic behavior of observables in the large-scale limit. Our approach does not rely on the ball averaging operator or Nevo's ergodic theorem. Instead, we introduce a new method based on the hyperbolic wave equation and the quantitative exponential mixing of the geodesic flow established by Ratner and Matheus.