Critical exponents on hyperbolic surfaces with long boundaries and the asymptotic Weil-Petersson form

2026/06/02 by Henry Talbott
Mathematics · Computer Science · #Geometric Analysis and Curvature Flows #advanced mathematical theories #Advanced Mathematical Modeling in Engineering

paper · doi:10.1016/j.aim.2026.111061

Abstract

We study the critical exponent random variable δ_X on moduli spaces of hyperbolic surfaces with boundary, using the normalized Weil-Petersson measures dμ_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 δ_Γ on moduli spaces of metric ribbon graphs with the normalized Kontsevich measures dμ_K, proving an asymptotic convergence-in-mean result in the long boundary length regime. In particular, we show that dμ_K approximately pulls back to dμ_WP with quantitative uniform estimates.

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