Random infinite ideal angled graphs and ideal hyperbolic polyhedra

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.