Year: 2026
Type: preprint
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 $$ \mathbb E[L_f(P)]=2π-\fracπ{3}\mathbb E[deg(f)]. $$ 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.
Source: arXiv (Cornell University)
Authors: Ge, Huabin, Lu, Yangxiang, Wang, Chuwen, Zhou, Tian
Institutions: Renmin University of China, Peking University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172536229