Geometry of Unimodular Random Hyperbolic Polyhedra

Abstract

We develop a geometric and probabilistic theory of unimodular random infinite trivalent hyperbolic polyhedra in ${\mathbb{H}}^3$, allowing ordinary, ideal, and hyperideal vertices. Our results relate local dihedral geometry to global conformal type and random-walk behavior. A central role is played by a geometric characteristic $L_f(P)$, determined by the dihedral angles incident to a face $f$. We establish the unimodular Gauss-Bonnet identity $$\mathbb{E}[L_f(P)]=2\pi -\frac{\pi }{3}\mathbb{E}[deg(f)].$$ Together with circle-pattern uniformization, this yields a sharp parabolic-hyperbolic dichotomy governed by the expected geometric characteristic. We also introduce polyhedral soficity and prove that admissible Benjamini-Schramm limits of finite trivalent hyperbolic polyhedra are parabolic. We further establish a quantitative Ring Lemma for regular circle patterns, providing degree-sensitive geometric control without assuming bounded degrees. Combined with unimodular probability methods, this estimate enables us to identify the Poisson boundary of the face random walk and establish positive linear hyperbolic speed. Our results extend the geometric-characteristic theory beyond ideal polyhedra, connecting three-dimensional geometry, unimodular Gauss-Bonnet theory, and stochastic boundary behavior.

arXiv commentv2: We have revised the title, abstract, and introduction to improve the clarity and presentation of the paper, and corrected several typographical errors throughout the manuscript