Thick trace at infinity for the Hyperbolic Radial Spanning Tree

TL;DR

This paper proves that in the hyperbolic Radial Spanning Tree, no infinite subtree can have all paths sharing the same asymptotic direction, and any infinite subtree's directions cover a positive measure set, indicating a 'thick' trace at infinity.

Contribution

It establishes the absence of 'very exceptional' directions with multiple infinite paths in the hyperbolic RST and shows all infinite subtrees have a positive measure of asymptotic directions.

Findings

No three infinite paths share the same asymptotic direction in 2D hyperbolic RST.

Any infinite subtree's asymptotic directions form a set of positive measure.

The hyperbolic RST contains no 'thin' infinite subtrees with a single asymptotic direction.

Abstract

Since the works of Howard and Newman (2001), it is known that in straight radial rooted trees, with probability 1, infinite paths all have an asymptotic direction and each asymptotic direction is reached by (at least) an infinite path. Moreover, there exists a set of 'exceptionnal' directions reached by (at least) two infinite paths which is random, dense and only countable in dimension 2. Howard and Newman's method says nothing about (random) directions reached by more than two infinite paths and, in particular, if such 'very exceptionnal' directions exist in dimension 2. In this paper, we prove that the answer is no for the hyperbolic Radial Spanning Tree (RST): in dimension 2, this tree does not contain 3 infinite paths with the same (random) asymptotic direction with probability one. Turned in another way, this means that there is no infinite but thin subtree in the hyperbolic RST, i.e. whose infinite paths would all have the same asymptotic direction. We actually prove a stronger result in dimension $d+1$, $d\geq 1$, stating that any infinite subtree of the hyperbolic RST a.s. generates a thick trace at infinity, i.e. the set of asymptotic directions reached by its infinite paths has a positive measure.