标题： 随机穿孔双曲曲面与布朗运动球 显示英文标题

标题： Random punctured hyperbolic surfaces & the Brownian sphere

摘要： 我们考虑随机的0亏格双曲曲面$\mathcal{S}_n$，带有$n + 1$个尖点，根据Weil-Petersson测度进行采样。 我们证明，在将度量按$n^{-1/4}$进行缩放后，曲面$\mathcal{S}_n$在分布上收敛到布朗球——一个与2-球同胚的随机紧致度量空间，表现出分形几何，并作为各种随机平面图模型中的普遍标度极限出现。 在不缩放度量的情况下，我们建立了$\mathcal{S}_n$对一个随机无限体积双曲曲面的局部Benjamini--Schramm收敛，该曲面有可数多个尖点，与$\mathbb{R}^2 \setminus \mathbb{Z}^2$同胚。 我们的证明借鉴了随机平面图理论中的技术。 特别是，我们通过一组带有连续标签的平面树对带尖点的双曲曲面进行了编码，类似于Schaeffer的双射。 这种编码源自Epstein-Penner分解，并通过一系列变换简化为单类型Galton--Watson树的模型，从而可以应用已知的不变性原理。 显示英文摘要

摘要： 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.