Random punctured hyperbolic surfaces & the Brownian sphere
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914006139142144 |
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| author | Budd, Timothy Curien, Nicolas |
| author_facet | Budd, Timothy Curien, Nicolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18792 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random punctured hyperbolic surfaces & the Brownian sphere Budd, Timothy Curien, Nicolas Probability Mathematical Physics Geometric Topology 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. |
| title | Random punctured hyperbolic surfaces & the Brownian sphere |
| topic | Probability Mathematical Physics Geometric Topology |
| url | https://arxiv.org/abs/2508.18792 |