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

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 ${\delta }_{\mathrm{\Gamma }}$ 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.