Random Genus-0 Hyperbolic Surfaces

• Random genus-0 hyperbolic surfaces are 2D Riemannian manifolds with constant negative curvature, topologically a sphere with multiple cusps.
• They are sampled from moduli space via the Weil–Petersson measure and are encoded using decorated plane binary trees capturing both geometric and combinatorial data.
• Global scaling limits converge to the Brownian sphere while local limits exhibit Benjamini–Schramm convergence, linking these surfaces to universal random planar map models.

Random genus-0 hyperbolic surfaces are two-dimensional Riemannian manifolds with constant negative curvature ($-1$), topologically the sphere with $n+1$ punctures (cusps). When equipped with a hyperbolic metric and sampled randomly from moduli space using the Weil–Petersson measure, these surfaces exhibit rich geometric and probabilistic properties, deeply intertwined with structures arising in random planar maps and universal scaling limits such as the Brownian sphere. Although the classical theory of hyperbolic surfaces concentrates on higher-genus cases, genus-0 surfaces with many punctures form a central object in the paper of random geometry, moduli spaces, combinatorial encodings, and scaling limits.

1. Definition and Structure of Genus-0 Hyperbolic Surfaces

A genus-0 hyperbolic surface with $n+1$ punctures is formally a point in the moduli space $\mathcal{M}_{0,n+1}$ of marked hyperbolic metrics on the sphere with $(n+1)$ cusps. These punctures act as ideal points at infinity, where the surface is completed by adding cusp neighborhoods determined by decorations—horocycle choices at each puncture.

Key structural features:

• The hyperbolic metric is unique up to isometries fixing the punctures and chosen decorations.
• The decorated Teichmüller space parametrizes such surfaces via the lengths, angles, or lambda-lengths associated with ideal triangulations.
• The Epstein–Penner decomposition associates to each surface a canonical cellulation with spine, giving rise to a combinatorial planar tree structure where edges correspond to ideal arcs between punctures.

For large $n$, these surfaces become increasingly “complex” in terms of their combinatorial decompositions, and their geometry is encoded efficiently by labeled plane trees that encapsulate both the topology and the metric data through associated continuous labels (such as allowed angle assignments at the tree vertices).

2. Weil–Petersson Probability Measure and Moduli Space

The Weil–Petersson (WP) measure is the canonical symplectic volume measure on the moduli space $\mathcal{M}_{0,n+1}$, derived from the WP two-form on Teichmüller space. For genus-0 hyperbolic surfaces with decorations, this measure is expressed in multiple coordinate systems:

• Using lambda-length coordinates for the edges of the ideal triangulation, the WP volume form becomes a product of Lebesgue measures (after suitable normalization and change of variables).
• For angle coordinates $(\theta_i)$ in a Euclidean triangle modeling an ideal hyperbolic triangle, the relation $\theta_1 + \theta_2 + \theta_3 = 2\pi$ constrains the allowed assignments, and the WP measure pushes forward to $2^{n-2}$ times Lebesgue measure on the space of binary trees with allowed angle assignments.

This probabilistic structure facilitates the sampling of random hyperbolic surfaces and underpins the derivation of geometric and combinatorial expectations—such as counting geodesics, volumes, and curves—in both global and local contexts. Notably, the WP measure is pivotal in establishing convergence to universal limiting objects.

3. Encoding Hyperbolic Surfaces via Plane Trees

A central advance in the paper of random genus-0 hyperbolic surfaces is the encoding of each surface as a plane binary tree with continuous labels, mirroring the paradigm of Schaeffer's bijection for quadrangulations:

• The Epstein–Penner spine yields a full binary tree in generic cases (with all non-origin horocycles degenerated).
• Each interior vertex of the tree encodes a hyperbolic triangle; the metric data is encapsulated via lambda-lengths $(\lambda_i)$, related to the angles $(\theta_i)$ as $\theta_1 + \theta_2 + \theta_3 = 2\pi$, under additional Delaunay conditions for local geometric consistency.
• The labeled tree decomposition supports a reduction to Galton–Watson branching processes, allowing the application of probabilistic invariance principles relevant to scaling limits and local geometry.

This encoding is not only bijective for generic surfaces but also carries the push-forward WP measure, enabling enumeration and statistical analysis directly in terms of tree structures and facilitating the paper of universality and limit laws.

4. Global Scaling Limits: Convergence to the Brownian Sphere

A landmark result is the global scaling limit of random genus-0 hyperbolic surfaces with $n+1$ punctures:

• After rescaling the hyperbolic metric by $n^{-1/4}$ and removing canonical cusp neighborhoods, the compact core $\mathcal{S}_n^\circ$ converges in the Gromov–Hausdorff sense to the Brownian sphere as $n \to \infty$.
• The Brownian sphere is a random compact metric space almost surely homeomorphic to the 2-sphere, with fractal geometry (Hausdorff dimension 4), and is the universal limit in random planar map models.
• The convergence is expressed as

$$(\mathcal{S}_n^\circ, n^{-1/4} \cdot d_{\text{hyp}}) \to (\mathfrak{m}_\infty, c_{\text{WP}} D^*)$$

where $c_{\text{WP}} = \frac{2\pi}{\sqrt{3 c_0}}$ and $c_0$ is the first positive zero of $J_0$, the Bessel function.

The underlying proof proceeds by first encoding the surface as a labeled tree, then showing that the contour and label processes (properly rescaled) converge to the Brownian excursion and Brownian snake, respectively, which together construct the Brownian sphere via a continuum quotiented tree.

5. Local Limit Geometry: Benjamini–Schramm Convergence

In the local regime, random genus-0 hyperbolic surfaces exhibit Benjamini–Schramm convergence:

• Fixing a distinguished “root” point according to the hyperbolic mass measure and taking $n \to \infty$, the pointed metric surface $(\mathcal{S}_n, \circ)$ converges locally to a unique infinite-volume hyperbolic surface $\mathcal{S}_\infty$.
• Topologically, $\mathcal{S}_\infty$ is homeomorphic to $\mathbb{R}^2 \setminus \mathbb{Z}^2$, i.e., the plane minus a countable dense set of cusps.
• In terms of tree encodings, the local structure corresponds to the limit objects from branching processes, such as the Kesten tree or Aldous’ sin-tree, reflecting the infinite nature of the limiting surface.

This convergence provides a precise model for the local geometry around typical points in large random genus-0 hyperbolic surfaces and serves as the hyperbolic analog of the Uniform Infinite Planar Triangulation (UIPT).

6. Geometric and Combinatorial Implications

The paper of random genus-0 hyperbolic surfaces via these probabilistic and combinatorial methods yields several significant implications:

• The link to the Brownian sphere situates random WP-sampled genus-0 hyperbolic surfaces within the same universality class as random planar maps, directly impacting 2D quantum gravity and statistical physics.
• Combinatorial encodings enable the application of generating function techniques and recurrence relations (e.g., Zograf’s formula for Weil–Petersson volumes) via tree enumeration and Bessel function analysis.
• The robustness of tree-based encodings and WP measure pushes opens avenues for studying higher-genus surfaces, systolic properties, the length spectrum, and spectral theory using methods from random planar geometry.

This framework highlights how ideas from moduli space, hyperbolic geometry, and probabilistic combinatorics coalesce in the paper of random genus-0 hyperbolic surfaces to reveal both global and local universality phenomena, intricate geometric regularity, and correspondences with enumeration and limit theory central to modern geometric analysis.