The Radial Spanning Tree is straight in all dimensions

TL;DR

The paper proves that the Radial Spanning Tree (RST) is almost surely straight in every dimension $d\ge 2$, extending the known planar result to higher dimensions. It introduces a renewal-type decomposition of RST paths and a symmetrization framework inspired by the Directed Spanning Forest to overcome intricate dependencies inherent in the radial Poisson setting. By combining this decomposition with classical concentration inequalities, the authors show that RST paths deviate negligibly from straight lines, yielding consequences for infinite branches, including asymptotic directions and omnidirectionality. The findings provide a robust, non-planar methodology for analyzing infinite geodesics in random geometric trees and deepen the understanding of straightness beyond two dimensions.

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.

Figures (9)

• Figure 1: Simulations of the RST within a finite ball around the origin, depicted as a dot. The picture on the left corresponds to $d=2$, while the picture on the right corresponds to $d=3$, with line color and thickness chosen to help visualizing the depth.
• Figure 2: Simulation of the planar RST in a ball centered at the origin. Edges appear in light gray, with thickness decreasing with length. In red are highlighted some sub-trees exiting the displayed area, contained within translucent red cones pointing toward their sub-roots, with arbitrarily chosen apertures that appear to decrease to $0$ with the distance to the origin, suggesting straightness. In black are highlighted some finite sub-trees remaining within the area, illustrating that RST sub-trees are not necessarily infinite.
• Figure 3: Visualization of the geometrical constraints imposed during the construction of the RST in dimension $2$. In translucent colors are represented the radial cropped balls that must be empty of points.
• Figure 4: Illustration of Theorem \ref{['thm_deviation_control']}.
• Figure 5: Illustration of Lemma \ref{['lemma_empty_ball']}. In green the worst setup possible for $B(c,\rho)$, with $\|c\|=1$ and $\rho=\ell=\|c+e_d\|$.

Theorems & Definitions (50)

• Proposition 1.1: Proposition 2.8 in howard2001geodesics
• Theorem 1.2
• Proposition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm_main']}
• Theorem 3.1
• Lemma 3.2
• proof