Random infinite ideal angled graphs and ideal hyperbolic polyhedra

TL;DR

This paper develops a comprehensive boundary and uniformization theory for random infinite ideal hyperbolic polyhedra and their dual graphs, extending previous results to more general cellular decompositions.

Contribution

It introduces a dichotomy theorem for unimodular random ideal angled graphs and extends boundary theory beyond triangulations to cellular decompositions.

Findings

Graph is a.s. ICP-parabolic iff expected sum of angles equals 2π

Random walk converges to boundary with positive hyperbolic speed in hyperbolic case

Boundaries (geometric, Poisson, Martin, Gromov) coincide in the studied setting

Abstract

This article aims to develop the uniformization and boundary theory of random infinite ideal hyperbolic polyhedra (abbr. IHP) and their dual 1-skeleton, i.e., ideal angled graphs (abbr. IAG) from multiple perspectives, including combinatorics, geometry, analysis and random walks. For unimodular random IAG, we establish an ICP analog of the dichotomy theorem of Angel-Hutchcroft-Nachmias-Ray [4,5]. Specifically, the character $T(\rho):=\sum_{e\ni\rho}\Theta_e$ of an IAG, introduced in [40], determines its ICP type: the graph is a.s. ICP-parabolic if and only if $\mathbb{E}[T(\rho)]=2\pi$. In the ICP-hyperbolic case, the simple random walk converges a.s. to $\partial\mathbb{D}$ with positive hyperbolic speed. Moreover, the geometric, Poisson, Martin, and Gromov boundaries coincide, extending the boundary theory of Angel-Barlow-Gurevich-Nachmias [3] and Hutchcroft-Peres [37] beyond triangulations to cellular decompositions. As a corollary of the aforementioned IHP/IAG duality, we obtain the systematic characterizations of the random IHP. To develop our theory, we strengthen and refine the Ring Lemma of Ge-Yu-Zhou [27] for ICP, which provides quantitative local control of the packing geometry. This key estimate makes it possible to extend the boundary theory beyond triangulations.