Random infinite ideal angled graphs and ideal hyperbolic polyhedra

This article aims to develop the theory of random infinite ideal hyperbolic polyhedra (abbr. IHP) from multiple perspectives, including combinatorics, geometry, analysis, and random walks. Our starting point is the one-to-one correspondence between IHP and ideal circle packings (ICP), which allows us to translate the theory of IHP into the language of ICP. We then extend the theories of Angel-Hutchcroft-Nachmias-Ray \cite{AHNR16,map} to the ICP setting. This extension is far from straightforward: the presence of dihedral angles introduces substantial new difficulties, requiring new estimates, techniques, and theoretical tools. In particular, we introduce a geometric characteristic number that provides a precise and effective characterization of infinite hyperbolic polyhedra.
An IHP $\mathcal P$ corresponds to a weighted planar infinite graph $(G,\Theta)$, called an ideal angled graph (abbr. IAG). For unimodular random IAG, we establish an ICP analog of the dichotomy theorem of Angel-Hutchcroft-Nachmias-Ray \cite{AHNR16,map}. Specifically, the geometric characteristic number $T(\rho)=2\pi-\sum_{e\ni\rho}\Theta_e$ of an IAG determines its ICP type: the graph is a.s. ICP-parabolic iff $\mathbb{E}[T(\rho)]=0$. 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 boundaries coincide, extending the boundary theory of Angel-Barlow-Gurevich-Nachmias \cite{ABGN16} and Hutchcroft-Peres \cite{HP17} beyond triangulations to cellular decompositions. As a corollary of the aforementioned IHP/IAG duality, we obtain systematic characterizations of random IHPs.