A Moebius inversion formula to discard tangled hyperbolic surfaces (2401.01601v1)
Abstract: Recent literature on Weil-Petersson random hyperbolic surfaces has met a consistent obstacle: the necessity to condition the model, prohibiting certain rare geometric patterns (which we call tangles), such as short closed geodesics or embedded surfaces of short boundary length. The main result of this article is a Moebius inversion formula, allowing to integrate the indicator function of the set of tangle-free surfaces in a systematic, tractable way. It is inspired by a key step of Friedman's celebrated proof of Alon's conjecture. We further prove that our tangle-free hypothesis significantly reduces the number of local topological types of short geodesics, replacing the exponential proliferation observed on tangled surfaces by a polynomial growth.
- Friedman-ramanujan functions and average counting of geodesics on random hyperbolic surfaces of large genus. in preparation, 2023.
- Friedman-ramanujan functions in random hyperbolic geometry and application to spectral gaps. arxiv.org/abs/2304.02678, 2023.
- Spectral gap of random hyperbolic surfaces. in preparation, 2023.
- Charles Bordenave. A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts. Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 53(6):1393–1439, 2020.
- Peter Buser. Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser, Boston, 1992.
- Travaux de Thurston sur les surfaces. Société Mathématique de France, Paris, 1991. Séminaire Orsay, Reprint of Travaux de Thurston sur les surfaces, Soc. Math. France, Paris, 1979 [MR0568308 (82m:57003)], Astérisque No. 66-67 (1991).
- Joel Friedman. A proof of Alon’s second eigenvalue conjecture. Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 720–724, 2003.
- On the relativized alon eigenvalue conjecture ii: Asymptotic expansion theorems for walks. arXiv:1911.05705, 2019.
- Making Curves Minimally Crossing by Reidemeister Moves. Journal of Combinatorial Theory, Ser. B, 70:134–156, 1997.
- Determinants of laplacians on random hyperbolic surfaces. ccepted by Comm. Pure and Applied Math., 2023, 2021.
- Towards optimal spectral gaps in large genus. arXiv:2103.07496, 2021.
- A dilogarithm identity on moduli spaces of curves. J. Differential Geom., 97(2):255–274, 2014.
- A random cover of a compact hyperbolic surface has relative spectral gap 316−ϵ316italic-ϵ\frac{3}{16}-\epsilondivide start_ARG 3 end_ARG start_ARG 16 end_ARG - italic_ϵ. Geometric and Functional Analysis, 32(3):595–661, 2022.
- Maryam Mirzakhani. Growth of Weil–Petersson volumes and random hyperbolic surfaces of large genus. Journal of Differential Geometry, 94(2):267–300, 2013.
- The tangle-free hypothesis on random hyperbolic surfaces. International Mathematics Research Notices, rnab160, 2021.
- Frédéric Naud. Determinants of laplacians on random hyperbolic surfaces. arxiv.org/abs/2301.09965, 2023.
- Random hyperbolic surfaces of large genus have first eigenvalues greater than 316−ϵ316italic-ϵ\frac{3}{16}-\epsilondivide start_ARG 3 end_ARG start_ARG 16 end_ARG - italic_ϵ. Geometric and Functional Analysis, 32(2):340–410, 2022.
Collections
Sign up for free to add this paper to one or more collections.
Paper Prompts
Sign up for free to create and run paper prompts using GPT-5.