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

TL;DR

This work develops a precise link between the critical exponent distributions of hyperbolic surfaces with boundary and those of metric ribbon graphs, via the spine map $\Phi$ and its long-boundary scaling. By combining spine-based geometry, topological recursion for volumes, and twist/dehn-twist analysis, the authors prove that, in the long-boundary regime, the rescaled surface exponent $\alpha\,\delta_X$ pushed forward along $\Phi_{1/\alpha}$, corrected by the Radon–Nikodym factor relating $d\mu_{WP}$ and $d\mu_K$, converges in mean to the graph exponent $\delta_\Gamma$, with an explicit Wasserstein rate $O(\alpha^{-1/12})$. They establish quantitative comparisons of the Weil–Petersson and Kontsevich forms, showing that $\Phi^*dV_K$ is absolutely continuous with respect to $dV_{WP}$ and that the associated coordinate changes are well-controlled in great and good regimes. The results yield distribution convergence for critical exponents and provide computable guidance for approximating surface exponent statistics via the combinatorial graph model, with potential impact on computing volume asymptotics and understanding length-spectrum universality in large-boundary limits.}

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.

Figures (16)

• Figure 1: An approximation for the distribution of critical exponent values on $MRG_{1,1}(1)$, based on smoothed data obtained from a computer simulation. Notice the lower bound of $3\log(2)\approx 2.08$.
• Figure 2: Examples of constructions used in this paper. Top left: a surface $X$ with $g=1$ and $n=1$ (i.e., a one-holed torus). Top right: the same surface with its spine graph marked. Notice that this metric ribbon graph has a single face. Bottom left: the same surface, with its ribs added in red and the three corridors shaded. Bottom right: the same surface, with its intercostals added in magenta and its two sectors shaded.
• Figure 3: A graph of $\delta_\Gamma$ as $\Gamma$ varies over the domain $F_{x,y}$ defined above.
• Figure 4: Four possible ways two arcs may overlap on a trivalent ribbon graph. We consider the left two to represent one intersection each, and the right two to not represent intersections. Note that the cases may be distinguished by examining the cyclic edge orderings at the vertices $v_1$ and $v_2$.
• Figure 5: Two possible cases of constructing a pair of pants from a given edge (blue). In the first case, the pants has two boundaries (red) that are also boundaries of the surface, and one boundary that is a geodesic on the surface (green). In the second case, the pants has one boundary that is also a boundary of the surface (red) and two boundaries that are geodesics on the surface (green).

Theorems & Definitions (83)

• Theorem 1.1: Selberg, AS56, Lalley, SL89, Cor. 11.2
• Remark 1.2
• Remark 1.3
• Theorem 1.4: Main result, non-technical version
• Remark 1.5
• Theorem 1.6
• Corollary 1.7
• Corollary 1.8
• Lemma 1.9
• Definition 1.10