Unimodular Hyperbolic Triangulations: Circle Packing and Random Walk

TL;DR

This paper characterizes the circle packing type of unimodular random plane triangulations, linking parabolicity to amenability and expected degree, and explores properties of random walks in hyperbolic triangulations including convergence, boundary behavior, and positive speed.

Contribution

It provides a new characterization of circle packing types in unimodular triangulations and analyzes random walk behavior in hyperbolic cases, including convergence and boundary properties.

Findings

Parabolic type corresponds to amenability and expected degree six.

In hyperbolic cases, random walks converge to the boundary with full support.

Random walks in hyperbolic triangulations have positive speed.

Abstract

We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons. As a part of this, we obtain an alternative proof of the Benjamini-Schramm Recurrence Theorem. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.