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Critical Exponents on Hyperbolic Surfaces with Long Bound...
[Submitted on 14 Jan 2025 (v1), last revised 24 Aug 2026 (this v · 2025-01-15 · via math updates on arXiv.org

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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.

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)