10. October 2025

Weil-Petersson random ideal hyperbolic polyhedra

Figure A random ideal hyperbolic polyhedron with 165 vertices sampled from the Weil-Petersson measure on the associated moduli space of punctured hyperbolic surfaces.

Figure Random ideal polyhedra with increasing number of vertices: 4, 5, 6, ..., 18.

Figure Random ideal polyhedra with increasing number of vertices: 19, 20, 21, ..., 38.

Video A rotating random ideal polyhedron with 78 vertices shown in the Poincare ball model of hyperbolic space.

Video A rotating random ideal polyhedron with 522 vertices shown in the Poincare ball model of hyperbolic space.

Mathematical background

The \(n\)-punctured sphere can be equipped with a unique complete hyperbolic metric conformal to the standard sphere metric, in which the neighbourhoods of the punctures become infinitely long cusps. Because of this it is difficult to faithfully visualize the hyperbolic metric by embedding in three-dimensional Euclidean space. However, by work of Igor Rivin [1], it can be uniquely realized as the boundary geometry of an ideal hyperbolic polyhedron in three-dimensional hyperbolic space, where “ideal” means that the vertices of the polyhedron are all on the boundary at infinity. This image shows the three-dimensional hyperbolic space displayed in the Poincaré disk/ball model, such that the vertices are on the sphere.

The moduli space of hyperbolic metrics on the n-punctured sphere has dimension \(2n-6\), because it can be parametrized by the positions of \(n-3\) of the punctures on the sphere (the position of 3 punctures can be fixed due to Moebius symmetry). This moduli space comes equipped with a natural measure determined by its Weil-Petersson symplectic structure. It’s a nice fact that the total measure is finite, so that upon normalization it can be thought of as a probability measure. The corresponding random hyperbolic metric is called the “Weil-Petersson random \(n\)-cusped sphere”. Such random surfaces, particularly their generalization to higher genus, have been subject of many papers at the interface between geometry and probability, starting with the foundational works of Maryam Mirzakhani [2]. The images and videos show samples of the Weil-Petersson random surface for fixed number \(n\) of cusps.

In the recent work [3] it was shown that upon rescaling the metric of the Weil-Petersson random \(n\)-cusped sphere by \(n^{-1/4}\) it converges to Brownian sphere.

References

[1] Rivin, Igor. Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space. Lecture Notes in Pure and Applied Mathematics (1994): 275-275.

[2] Mirzakhani, Maryam. Growth of Weil-Petersson volumes and random hyperbolic surface of large genus. Journal of Differential Geometry, 94, no. 2 (2013), pp.267-300.

[3] Budd, Timothy, and Nicolas Curien. Random punctured hyperbolic surfaces & the Brownian sphere. arXiv preprint arXiv:2508.18792 (2025).