Hyperbolic Radial Spanning Tree

TL;DR

This paper extends the Radial Spanning Tree to hyperbolic spaces of any dimension, analyzing its infinite branches and asymptotic directions using a novel approach involving hyperbolic Directed Spanning Forests.

Contribution

It introduces a new hyperbolic RST model, extending Euclidean properties to hyperbolic space and developing a unique method to analyze infinite branches in higher dimensions.

Findings

Almost surely, every infinite branch has an asymptotic direction.

Each asymptotic direction is reached by at least one infinite branch.

The branch converging to any deterministic asymptotic direction is unique almost surely.

Abstract

We define and analyze an extension to the $d$-dimensional hyperbolic space of the Radial Spanning Tree (RST) introduced by Baccelli and Bordenave in the two-dimensional Euclidean space (2007). In particular, we will focus on the description of the infinite branches of the tree. The properties of the two-dimensional Euclidean RST are extended to the hyperbolic case in every dimension: almost surely, every infinite branch admits an asymptotic direction and each asymptotic direction is reached by at least one infinite branch. Moreover, the branch converging to any deterministic asymptotic direction is unique almost surely. To obtain results for any dimension, a completely new approach is considered here. \tvc{Our strategy mainly involves the two following ingredients, that rely on the hyperbolic Directed Spanning Forest (DSF) introduced and studied in Flammant (2019).} First, the hyperbolic metric allows us to obtain fine control of the branches' fluctuations in the hyperbolic DSF without using planarity arguments. Then, we couple the hyperbolic RST with the hyperbolic DSF and conclude.