The $\ell_{\infty}$ Directed Spanning Forest

TL;DR

The paper analyzes the ℓ_{ finfinity}-directed spanning forest built on a homogeneous Poisson point process in the plane, proving almost sure connectivity and deriving a sharp tail bound for the coalescing time of two DSF paths. It introduces a joint exploration framework and a sequence of renewal steps to control the history regions generated by the paths, with distinct constructions for a single path (k=1) and for multiple paths (k≥2). The main result shows P(T>t) ≤ C_0 / √t, mirroring the diffusion-like scaling observed in the ℓ_2 case but requiring new geometric arguments due to the ℓ_{ finfinity} metric. The work lays groundwork toward convergence to the Brownian web under diffusive scaling and extends techniques from prior ℓ_2 analyses to the ℓ_{ finfinity} setting.

Abstract

We study the $\ell_{\infty}$\textit{ directed spanning forest}(DSF), which is a directed forest with vertex set given by a homogeneous Poisson point process such that each Poisson point connects to the nearest Poisson point (in $\ell_{\infty}$ distance) with a strictly larger $y$-coordinate. In this paper, we prove that the $\ell_{\infty}$ DSF is connected and we find optimal estimates on the tail distribution of coalescing time of two $\ell_{\infty}$ DSF paths. Similar estimates were earlier obtained in \cite{coupier20212d} for the $\ell_2$ (Euclidean) DSF and showed that when properly scaled, it converges in distribution to the Brownian web. The geometry of $\ell_\infty$ balls compel us to develop new argument.

Figures (5)

• Figure 1: This picture shows the first $5$ steps of the joint exploration process $\{ g_n( \mathbf{x}^1), g_n( \mathbf{x}^2), g_n( \mathbf{x}^3)\}_{n \ge 0}$ starting from $\mathbf{x}^1, \mathbf{x}^2, \mathbf{x}^3$ ( denoted by red dots). The point $\mathbf{x}^1$ moves first, i.e $W^{\text{move}}_0 = \{ \mathbf{x}^1 \}$ and on the next step $\mathbf{x}^2$ moves. The fifth step is the first time that the point $\mathbf{x}^3$ moves implying $W^{\text{move}}_5 = \{ \mathbf{x}^3 \}$ and $W^{\text{stay}}_5 = \{g_5( \mathbf{x}^1) , g_5( \mathbf{x}^2) \}$. The grey region represents the history set $H_5( \mathbf{x}^1, \mathbf{x}^2, \mathbf{x}^3)$.
• Figure 2: This picture is an illustration of the renewal step for the marginal process $\{g_n( \mathbf{x}) : n \geq 0\}$. The grey region represents the history set $H_3$. On the fourth step the point $g_3( \mathbf{x})$ connects to the top boundary and we also have $H_4 = \emptyset$, i.e, the renewal event occurs.
• Figure 3: An illustration of top step. Blue part denotes $\rho^+_{T, n+1}$ and union of blue part and red part denote $\rho^+_{ n+1}$. The grey region represents $H^\text{new}_{n+1} = B^+(g_n(\mathbf{x}^1), ||g_n(\mathbf{x}^1) - g_{n+1}(\mathbf{x}^1)||_\infty )$.
• Figure 4: This picture is an illustration of the idea of proof of Lemma \ref{['lem:ubound_Lprocess_1']}. The point $g_n(\mathbf{x}^1)$ connects with a Poisson point in the region $\Box^r_n \cup \Box^r_n$ rather than connecting with a Poisson point in the region $g_n^\uparrow \circ [ - L_n - 1, L_n + 1] \times [0,1]$. As a result, the event $E_{ n + 1}$ occurs, and the height of the history set can increase by at most 1.
• Figure 5: This picture represents joint renewal step of the joint exploration process $\{g_n(\mathbf{x}^1), g_n(\mathbf{x}^2), H_n): n \ge 0 \}$ with $k=2$ trajectories. Red dots represent projected vertices $g^{\uparrow, 1}_{\tau_1}$ and $g^{\uparrow, 2}_{\tau_1}$, i.e., positions of restart. The grey region denotes the history set $H_{\tau_1}( \mathbf{x}^1, \mathbf{x}^2)$.

Theorems & Definitions (35)

• Theorem 1
• Proposition 1
• Proposition 2
• Corollary 1
• Definition 1: 'Top' step and 'Up' step for $k=1$
• Remark 1
• Lemma 1
• proof
• Corollary 2
• Lemma 2