Geometry of Unimodular Random Hyperbolic Polyhedra
Abstract
We develop a geometric and probabilistic theory of unimodular random infinite trivalent hyperbolic polyhedra in โ3, 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 Lf(P), determined by the dihedral angles incident to a face f. We establish the unimodular Gauss-Bonnet identity ๐ผ[Lf(P)]=2ฯโฯ/3๐ผ[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.
Related
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BibTeX
@misc{ge2026geometry,
title = {{Geometry of Unimodular Random Hyperbolic Polyhedra}},
author = {Huabin Ge and Yangxiang Lu and Chuwen Wang and Tian Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.03575},
eprint = {2608.03575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03575},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Geometry of Unimodular Random Hyperbolic Polyhedra
AU - Huabin Ge
AU - Yangxiang Lu
AU - Chuwen Wang
AU - Tian Zhou
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.03575
UR - https://arxiv.org/abs/2608.03575
AB - We develop a geometric and probabilistic theory of unimodular random infinite trivalent hyperbolic polyhedra in โ^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 ๐ผ[L_f(P)]=2ฯโฯ/3๐ผ[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.
ER -