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Random infinite ideal angled graphs and ideal hyperbolic polyhedra

Huabin Ge, Yangxiang Lu, Chuwen Wang, Tian Zhou

TL;DR

The paper develops a unified framework connecting random infinite ideal hyperbolic polyhedra (IHP) with their dual ideal angled graphs (IAG). It formulates and proves an ICP–IHP correspondence and an IHP/IAG duality, then establishes a dichotomy for unimodular IAGs via the angle character $T(\rho)$, showing $\mathbb{E}[T(\rho)]=2\pi$ characterizes ICP-parabolicity and amenability, while $\mathbb{E}[T(\rho)]>2\pi$ yields ICP-hyperbolicity and non-amenability with recurrent versus transient behavior. The boundary theory is developed to identify the geometric, Poisson, Martin, and Gromov boundaries and to show non-atomic, full-support exit measures for hyperbolic cases, with a refined Ring Lemma enabling these results beyond triangulations. The work also proves convergence and speed results for random walks, including positive hyperbolic speed linked to radius decay, and surveys related topics such as unbounded degree scenarios, multi-ends, and combinatorial Ricci flow, situating the theory at the crossroads of unimodular random graphs, discrete conformal geometry, and hyperbolic polyhedral geometry.

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 $T(ρ):=\sum_{e\niρ}Θ_e$ of an IAG, introduced in [40], determines its ICP type: the graph is a.s. ICP-parabolic if and only if $\mathbb{E}[T(ρ)]=2π$. 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.

Random infinite ideal angled graphs and ideal hyperbolic polyhedra

TL;DR

The paper develops a unified framework connecting random infinite ideal hyperbolic polyhedra (IHP) with their dual ideal angled graphs (IAG). It formulates and proves an ICP–IHP correspondence and an IHP/IAG duality, then establishes a dichotomy for unimodular IAGs via the angle character , showing characterizes ICP-parabolicity and amenability, while yields ICP-hyperbolicity and non-amenability with recurrent versus transient behavior. The boundary theory is developed to identify the geometric, Poisson, Martin, and Gromov boundaries and to show non-atomic, full-support exit measures for hyperbolic cases, with a refined Ring Lemma enabling these results beyond triangulations. The work also proves convergence and speed results for random walks, including positive hyperbolic speed linked to radius decay, and surveys related topics such as unbounded degree scenarios, multi-ends, and combinatorial Ricci flow, situating the theory at the crossroads of unimodular random graphs, discrete conformal geometry, and hyperbolic polyhedral geometry.

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 of an IAG, introduced in [40], determines its ICP type: the graph is a.s. ICP-parabolic if and only if . In the ICP-hyperbolic case, the simple random walk converges a.s. to 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.
Paper Structure (30 sections, 34 theorems, 266 equations, 10 figures)

This paper contains 30 sections, 34 theorems, 266 equations, 10 figures.

Table of Contents

  1. Introduction
  2. $\mathrm{ICP}$-$\mathrm{IHP}$ correspondence and $\mathrm{IHP}/\mathrm{IAG}$ duality.
  3. Dichotomy of unimodular $\mathrm{IAG}$.
  4. Boundary theory for $\mathrm{IAG}$ and unimodular $\mathrm{IAG}$.
  5. Random infinite $\mathrm{IHP}$ theory.
  6. Background and related work.
  7. Preliminaries
  8. Ideal circle packing and ideal angeled graph
  9. Infinite ideal polyhedra in $\mathbb{H}^3$
  10. Unimodular random rooted graphs
  11. Vertex extremal length and random walks
  12. Invariant amenability
  13. Embeddings and boundaries
  14. Circle packing (CP) embedding.
  15. Ideal circle packing ($\mathrm{ICP}$) embedding.
  16. ...and 15 more sections

Key Result

Theorem 1.0

Let $\mathcal{D}=(V,E,F)$ be a finite cellular decomposition of the sphere $\mathbb{S}^{2}$ and let $\Theta\in(0,\pi)^{E}$. Then there exists an ideal polyhedron $\mathcal{P}$ which is combinatorially equivalent to the Poincaré dual of $\mathcal{D}$ with dihedral angle $\Theta(e^{*})=\Theta(e)$ if a Moreover, the ideal hyperbolic polyhedron is unique up to isometry.

Figures (10)

  • Figure 1: $\mathrm{ICP}$-$\mathrm{IHP}$ correspondence and $\mathrm{IHP}/\mathrm{IAG}$ duality.
  • Figure 2: Ideal circle packing
  • Figure 3: Quadrilateral
  • Figure 4: Mass transport $m(u,v)$
  • Figure 5: Mass transport between $\rho$ and $v_2$
  • ...and 5 more figures

Theorems & Definitions (71)

  • Theorem 1.0: Rivin Riv96
  • Remark
  • Definition 1.0: $\mathrm{IAG}$
  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark
  • ...and 61 more