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

TL;DR

This paper investigates the behavior of the critical exponent on hyperbolic surfaces with boundaries, demonstrating an asymptotic convergence between Weil-Petersson and Kontsevich measures in the long boundary length limit.

Contribution

It establishes a quantitative link between Weil-Petersson and Kontsevich measures, showing their asymptotic equivalence on moduli spaces of hyperbolic surfaces with boundaries.

Findings

Proves convergence-in-mean of critical exponents under different measures.

Provides uniform estimates for measure pullback.

Shows asymptotic equivalence in the long boundary length regime.

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.