Quantum Mixing and Benjamini-Schramm Convergence of Hyperbolic Surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917432135778304 |
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| author | Hippi, Kai |
| author_facet | Hippi, Kai |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_15504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Mixing and Benjamini-Schramm Convergence of Hyperbolic Surfaces Hippi, Kai Spectral Theory Mathematical Physics Dynamical Systems 81Q50, 37D40, 11F72 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. |
| title | Quantum Mixing and Benjamini-Schramm Convergence of Hyperbolic Surfaces |
| topic | Spectral Theory Mathematical Physics Dynamical Systems 81Q50, 37D40, 11F72 |
| url | https://arxiv.org/abs/2512.15504 |