Random punctured hyperbolic surfaces & the Brownian sphere

TL;DR

This paper demonstrates that random hyperbolic surfaces with punctures, when appropriately rescaled, converge to the Brownian sphere, revealing universal fractal geometry and connecting hyperbolic geometry with random planar map models.

Contribution

It introduces a novel encoding of punctured hyperbolic surfaces via labeled plane trees, facilitating the proof of convergence to the Brownian sphere.

Findings

Rescaled surfaces converge to the Brownian sphere.

Local limits are infinite-volume hyperbolic surfaces.

Encoding via plane trees enables application of invariance principles.

Abstract

We consider random genus-0 hyperbolic surfaces $\mathcal{S}_n$ with $n + 1$ punctures, sampled according to the Weil-Petersson measure. We show that, after rescaling the metric by $n^{-1/4}$, the surface $\mathcal{S}_n$ converges in distribution to the Brownian sphere - a random compact metric space homeomorphic to the 2-sphere, exhibiting fractal geometry and appearing as a universal scaling limit in various models of random planar maps. Without rescaling the metric, we establish a local Benjamini--Schramm convergence of $\mathcal{S}_n$ to a random infinite-volume hyperbolic surface with countably many punctures, homeomorphic to $\mathbb{R}^2 \setminus \mathbb{Z}^2$. Our proofs mirror techniques from the theory of random planar maps. In particular, we develop an encoding of punctured hyperbolic surfaces via a family of plane trees with continuous labels, akin to Schaeffer's bijection. This encoding stems from the Epstein-Penner decomposition and, through a series of transformations, reduces to a model of single-type Galton--Watson trees, enabling the application of known invariance principles.