Unimodular random ideal hyperbolic polyhedra: geometric characteristics and random walks

Abstract

We initiate a systematic study of unimodular random infinite ideal hyperbolic polyhedra in three-dimensional hyperbolic space, developing a geometric and probabilistic theory of their global behavior. At the heart of this theory is an intrinsic geometric characteristic, determined entirely by the dihedral angles surrounding a polyhedral face. We establish a unimodular Gauss-Bonnet identity and show that the expectation of this local geometric invariant completely determines the global conformal type: under suitable assumptions, a random ideal polyhedron is almost surely parabolic precisely when the expected characteristic vanishes, and hyperbolic precisely when it is negative. This provides a sharp geometric classification of random infinite ideal polyhedra. Building on the correspondence with ideal circle patterns, we further develop a refined Ring Lemma that yields quantitative geometric control without bounded-degree assumptions. Combining these geometric tools with unimodular probability methods, we identify the Poisson boundary of the face random walk and establish positive linear hyperbolic speed. Together, these results establish a local-to-global framework linking intrinsic dihedral geometry, conformal uniformization, and stochastic behavior at infinity for unimodular random infinite ideal hyperbolic polyhedra.

arXiv commentWe have revised the title, abstract, and introduction, and clarified several statements