Random hyperbolic polyhedra in horoballs

Abstract

We study the geodesic convex hull of a stationary Poisson point process restricted to a horoball in $d$-dimensional hyperbolic space. The resulting random set is an unbounded hyperbolic polyhedron with a distinguished ideal direction. Projecting its boundary facets to the bounding horosphere yields a stationary Euclidean tessellation of $\mathbb{R}^{d-1}$, which we identify as a dual Poisson--Laguerre tessellation with an explicit height density. We derive an exact formula for its cell intensity and, in the critical regime where the intensity of the Poisson point process is matched with the height of the horoball, we prove local convergence of the projected tessellation to the classical Poisson--Delaunay tessellation in $\mathbb{R}^{d-1}$. As consequences, the typical cell converges in distribution and the intensities of all $k$-dimensional faces converge to their Poisson--Delaunay counterparts. We also study a localised volume functional of the hyperbolic convex hull, determine its limiting expectation in the same regime, and use an Efron-type identity to obtain a second-order asymptotic expansion for the vertex intensity. In addition, we derive an exact formula for the expected localised surface area and determine its critical asymptotics. In dimensions $d\ge3$ the expected unnormalised local surface area converges to a finite limit, whereas in dimension $d=2$ it exhibits logarithmic growth.