Critical Exponents on Hyperbolic Surfaces with Long Boundaries and the Asymptotic Weil-Petersson Form

[2501.08447] Critical Exponents on Hyperbolic Surfaces with Long Boundaries and the Asymptotic Weil-Petersson Form Skip to main content Search Submit Donate Log in Mathematics > Geometric Topology arXiv:2501.08447 (math) [Submitted on 14 Jan 2025 ( v1 ), last revised 24 Aug 2026 (this version, v2)] Title: Critical Exponents on Hyperbolic Surfaces with Long Boundaries and the Asymptotic Weil-Petersson Form Authors: Henry Talbott View PDF HTML (experimental) Abstract: We study the critical exponent random variable $\delta_X$ on moduli spaces of hyperbolic surfaces with boundary, using the normalized Weil-Petersson measures $d\mu_{WP}$ as probability measures. We use the spine graph construction of Bowditch and Epstein to compare this random variable to the corresponding critical exponent random variable $\delta_\Gamma$ on moduli spaces of metric ribbon graphs with the normalized Kontsevich measures $d\mu_K$, proving an asymptotic convergence-in-mean result in the long boundary length regime. In particular, we show that $d\mu_K$ approximately pulls back to $d\mu_{WP}$ with quantitative uniform estimates. Comments: 64 pages, 16 figures. v2 Subjects: Geometric Topology (math.GT) ; Probability (math.PR) Cite as: arXiv:2501.08447 [math.GT] (or arXiv:2501.08447v2 [math.GT] for this version) https://doi.org/10.48550/arXiv.2501.08447 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Henry Talbott [ view email ] [v1] Tue, 14 Jan 2025 21:35:37 UTC (4,708 KB) [v2] Mon, 24 Aug 2026 18:06:57 UTC (5,546 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GT < prev | next > new | recent | 2025-01 Change to browse by: math math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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