Random punctured hyperbolic surfaces & the Brownian sphere

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 $n^{-1\mathrm{/}4}$, 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.