Infinite geodesics in hyperbolic random triangulations

TL;DR

This paper investigates the structure of infinite geodesics in hyperbolic random triangulations, revealing they form a Galton-Watson tree, and explores implications for boundary theory and hyperbolicity properties.

Contribution

It establishes that infinite geodesics in hyperbolic triangulations form a Galton-Watson tree and introduces a new boundary concept linked to the Poisson boundary.

Findings

Infinite geodesics form a supercritical Galton-Watson tree.

The tree of geodesics defines a new boundary as a Poisson boundary realization.

Hyperbolic triangulations exhibit weaker Gromov-hyperbolicity and have bi-infinite geodesics.

Abstract

We study the structure of infinite geodesics in the Planar Stochastic Hyperbolic Triangulations $\mathbb{T}_{\lambda}$, which are the hyperbolic analogs of the UIPT. We prove that these geodesics form a supercritical Galton--Watson tree with geometric offspring distribution. The tree of infinite geodesics in $\mathbb{T}_{\lambda}$ provides a new notion of boundary, which is a realization of the Poisson boundary. By scaling limits arguments, we also obtain a description of the tree of infinite geodesics in the hyperbolic Brownian plane. Finally, by combining our main result with a forthcoming paper, we obtain new hyperbolicity properties of $\mathbb{T}_{\lambda}$: it satisfies a weaker form of Gromov-hyperbolicity and admits bi-infinite geodesics.