The Radial Spanning Tree is straight in all dimensions

TL;DR

This paper proves that the Radial Spanning Tree (RST) is almost surely straight in all dimensions, extending previous planar results to higher dimensions using renewal decomposition and concentration inequalities.

Contribution

The paper introduces a new approach to prove the straightness of RST in any dimension, overcoming previous limitations related to planarity and complex dependencies.

Findings

RST is almost surely straight in all dimensions

Infinite branches are asymptotically directed

Directions of infinite branches form a dense subset

Abstract

The Radial Spanning Tree (RST) in dimension $d\geq2$ is a random geometric graph constructed on a homogeneous Poisson point process $\mathcal N$ in $\mathbb R^d$ augmented by the origin, with edges connecting each $x\in\mathcal N$ to the nearest point $y\in\mathcal N\cup\{0\}$ that lies closer to $0$ than $x$, with respect to the Euclidean distance. By construction, it forms almost surely a tree rooted at $0$. The RST was introduced in 2007 by Baccelli and Bordenave, who investigated straightness, a deterministic property introduced by Howard and Newman in 2001, to derive information about the asymptotic directions of infinite branches. They proved that the RST is almost surely straight in dimension $2$, which directly implies that all infinite branches are asymptotically directed, every possibility is attained, and directions reached by multiple infinite branches form a dense subset. However, their approach relies crucially on planarity, preventing any straightforward extension to higher dimensions. In this paper, we close this gap by proving that the RST is almost surely \emph{straight} in any dimension, thereby obtaining the same consequences for the behavior of infinite branches. Our approach resolves the key barriers in the study of the RST, notably those posed by the complex dependency structure combined with the radial nature of the model, and especially beyond the planar setting. It relies on tools developed for the analysis of the Directed Spanning Forest, a closely related model, including recent progress by the author in 2025. Specifically, a key contribution of this work is the construction of a suitable renewal-type decomposition of RST paths. Leveraging this decomposition together with classical concentration inequalities, we show that RST paths cannot deviate far from straight lines and derive straightness.