The Directed Spanning Forest: coalescence versus dimension

Abstract

For $p\in[1,\infty]$, the $\ell^p$ directed spanning forest (DSF) of dimension $d\geq 2$ is an oriented random geometric graph whose vertex set is given by a homogeneous Poisson point process $\mathcal N$ on $\mathbb R^d$ and whose edges consist of all pairs $(x, y)\in\mathcal N^2$ such that $y$ is the closest point to $x$ in $\mathcal N$ for the $\ell^p$ distance among points with a strictly larger $e_d$ coordinate. First introduced by Baccelli and Bordenave in 2007 in the case $p=d=2$, this graph has a natural forest structure. In this work, we study the number of disjoint trees in the $\ell^p$ DSF for arbitrary dimensions $d\geq2$ and various values of $p\in [1,\infty]$. We prove that for $p\in\{1, 2,\infty\}$, the graph is almost surely a tree when $d=3$, and consists of infinitely many disjoint trees when $d\geq 4$. Additionally, we show that for all $p\in[1,\infty]$, the DSF in dimension $2$ is almost surely a tree and, under appropriate diffusive scaling, converges weakly to the Brownian web, generalizing the result previously known for p=2. Although these results were expected from a heuristic point of view, and the main strategies and tools were largely understood, extending them beyond the planar setting ($d=2$) and to the singular case $p=\infty$ presented a significant challenge. Notably, in the absence of planarity, which plays a crucial role in existing arguments, delicate and innovative techniques were required to manage the complex geometric dependencies of the model. We develop substantially new ideas to handle arbitrary dimension $d\geq2$ and various values of $p\in [1,\infty]$ within a unified framework. In particular, we introduce a novel stochastic domination argument that allows us to compare the fully dependent model with a simplified version in which the geometric correlations are suppressed.