Supercritical causal maps : geodesics and simple random walk

TL;DR

This paper investigates supercritical Galton-Watson tree-based planar maps with horizontal connections, revealing hyperbolic properties, geodesic structures, and random walk behaviors, contributing to understanding random hyperbolic geometry.

Contribution

It introduces a new model of random hyperbolic maps derived from supercritical Galton-Watson trees and analyzes their geometric and probabilistic properties.

Findings

Maps admit bi-infinite geodesics

They satisfy a weak form of Gromov-hyperbolicity

Random walk has positive speed when the tree has no leaves

Abstract

We study the random planar maps obtained from supercritical Galton--Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and Nachmias in arXiv:1710.03137, and a natural model of random hyperbolic geometry. We first establish metric hyperbolicity properties of these maps: we show that they admit bi-infinite geodesics and satisfy a weak version of Gromov-hyperbolicity. We also study the simple random walk on these maps: we identify their Poisson boundary and, in the case where the underlying tree has no leaf, we prove that the random walk has positive speed. Some of the methods used here are robust, and allow us to obtain more general results about planar maps containing a supercritical Galton--Watson tree.