Diameter of a new model of random hyperbolic surfaces

TL;DR

The paper introduces a new random construction of closed hyperbolic surfaces by gluing pants according to a random 3-regular graph drawn via the configuration model, with iid edge parameters from a law $\nu$. It proves that the diameter grows logarithmically with genus, namely $\mathrm{diam}(\mathfrak S_{\nu,g}) \approx \frac{1}{\alpha_{\nu}} \ln g$, where $\alpha_{\nu}$ encodes the exponential growth rate of a corresponding tree-like surface, and provides concentration bounds for the associated ball-count process $N_R$. The authors develop a tree-based surrogate geometry $\widehat{\mathfrak S}_{\nu}$, establish subadditive and concentration techniques to obtain the growth rate $\alpha_{\nu}$, and transfer these results to the actual graph-based surfaces via a refined exploration argument. They also analyze the asymptotics of $\alpha_{\nu}$ as a function of pants boundary lengths and twist distributions, showing $\alpha_{l\otimes\nu_t} \to 1$ under uniform twists as $l\to\infty$, and discuss the implications for typical geometric behavior in large-genus random hyperbolic surfaces. The work connects random graph methods with hyperbolic geometry to illuminate the typical connectivity scale in randomly constructed surfaces.

Abstract

The study of random surfaces, especially in the asymptotics of large genus, has been of increasing interest in recent years. Many geometrical questions have analogous formulations in the theory of random graphs with a large number of vertices, and results obtained in one area can inspire the other. In this paper, we are interested in the diameter of random surfaces, a basic measure of the connectivity of the surface. We introduce a new class of models of random surfaces built from random graphs and we compute the asymptotics of the diameter of these surfaces, which is logarithmic in the genus of the surface. The strategy of the proof relies on a detailed study of an exploration process which is the analogue of the breadth-first search exploration of a random graph. Its analysis is based on subadditive and concentration techniques.

Figures (13)

• Figure 1.1: The correspondence between graphs and surfaces built with pair of pants.
• Figure 1.2: Two pairs of pants glued together with two different twists.
• Figure 1.3: An example of a 3-regular graph with 6 vertices obtained with the configuration model.
• Figure 2.1: The surface $\widehat{\mathfrak S}_\nu$, build from $\mathfrak T_3$, and in red, the surface $\mathfrak S_\nu$, obtained by only considering $\mathfrak B$.
• Figure 2.2: The region coloured on the right corresponds to the points at a distance less than $R$ from $\partial \rho^-$. The ball $\mathcal{B}_R$ is surrounded in blue. The squared vertices, in green, correspond to the pairs of pants in $\mathcal{S}_R$. In this example, $N_R = 4$.

Theorems & Definitions (76)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Theorem 2.5: Subgaussian concentration
• Definition 2.6