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arXiv · 2512.15504

Quantum Mixing and Benjamini-Schramm Convergence of Hyperbolic Surfaces

Kai Hippi
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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 Verdière, 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.

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Kai Hippi. 2026-04-24. Quantum Mixing and Benjamini-Schramm Convergence of Hyperbolic Surfaces. https://arxiv.org/abs/2512.15504

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