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Connected components and topological ends of stationary planar forests

Tom Garcia-Sanchez

Abstract

We study the topological structure of random geometric forests $G$ in the Euclidean plane under mild assumptions: non-crossing edges, stationarity, and finite edge intensity. The framework covers a broad range of constructions, including models based on stationary point processes as well as lattices, and encompasses many already well-studied examples among drainage networks, geodesic forests arising from first- and last-passage percolation, and minimal or uniform spanning trees. First, denoting by $N_k$ the number of $k$-ended connected components in $G$ for each $k\geq0$, we show that almost surely, all trees of $G$ have at most two topological ends, $N_0\in\{0,\infty\}$, $N_1\leq2$, and $N_1=2\implies N_2<\infty$. We then construct explicit examples realizing all possibilities compatible with these constraints, yielding a complete classification of the admissible topological structures for $G$. As a second result, we prove that under the additional assumptions that $G$ is non-empty, oriented, out-degree one, with all its directed paths going to infinity along a fixed deterministic direction, the situation reduces to a dichotomy: $G$ consists almost surely of either a unique one-ended tree, or infinitely many two-ended trees. Our proofs combine classical Burton-Keane type arguments with substantial new conceptual ideas using planar topology, resulting in a robust, unified approach.

Connected components and topological ends of stationary planar forests

Abstract

We study the topological structure of random geometric forests in the Euclidean plane under mild assumptions: non-crossing edges, stationarity, and finite edge intensity. The framework covers a broad range of constructions, including models based on stationary point processes as well as lattices, and encompasses many already well-studied examples among drainage networks, geodesic forests arising from first- and last-passage percolation, and minimal or uniform spanning trees. First, denoting by the number of -ended connected components in for each , we show that almost surely, all trees of have at most two topological ends, , , and . We then construct explicit examples realizing all possibilities compatible with these constraints, yielding a complete classification of the admissible topological structures for . As a second result, we prove that under the additional assumptions that is non-empty, oriented, out-degree one, with all its directed paths going to infinity along a fixed deterministic direction, the situation reduces to a dichotomy: consists almost surely of either a unique one-ended tree, or infinitely many two-ended trees. Our proofs combine classical Burton-Keane type arguments with substantial new conceptual ideas using planar topology, resulting in a robust, unified approach.

Paper Structure

This paper contains 6 sections, 25 theorems, 148 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Trifurcations
  3. Derivation of the dichotomy result
  4. Examples
  5. Corridor structure
  6. Proof of the planar ordering theorem

Key Result

Theorem I

With probability one, every connected component of $G$ is a tree with at most $2$ topological ends. Let $N_0, N_1$ and $N_2$ respectively denote the number of finite, one-ended and two-ended connected components of $G$, taking values in $\mathbb{Z}_+\cup\{\infty\}$. Then, $(N_0,N_1,N_2)$ is constant Moreover, this classification is optimal: every value for $(N_0,N_1,N_2)$ satisfying the above cons

Figures (4)

  • Figure 1: Illustration behind the proof of Proposition \ref{['prop_N1_leq_2']}. Dotted squares represent $c_{i, j}+(-k,k+\ell)^2$ and $c_{i', j'}+(-k,k+\ell)^2$ while bold lines represent edges of $\mathrm{Pend}(c_{i, j}+[0,k]^2)$ and $\mathrm{Pend}(c_{i', j'}+[0,k]^2)$. Here, $s=2$ and $\{\alpha_r'\}_{r\in\{1,2,3\}}\subset [\alpha_2,\alpha_3]$.
  • Figure 2: Simulation illustrating the proof of Lemma \ref{['lemma_bi_infinite_ray']} realized using the Broder-Aldous algorithm (see broder_aldous). The edges of $G_\mathrm{UST}$ and $G_\mathrm{UST}^*$ are represented by solid black and dotted gray lines, respectively, while those of $\mathrm{Cont}[G_\mathrm{UST}, \varepsilon]$ appear in red. Red and black dots indicate the vertices $\{w_1,w_1',w_2,w_2'\}$ and the points $\{z_1,z_1^*,z_2,z_2^*\}$, respectively. The translucent red region represents $\mathrm{Int} J$.
  • Figure 3: On the left, a simulation of $G_\mathrm{UST}^*$ and $G_\phi$ for some arbitrarily chosen increasing function $\phi:\mathbb{Z}_+\to\mathbb{Z}_+$. On the right, a simulation of $G_\mathrm{UST}$ with each edge $(v,v')$ colored according to $\max\{d_\mathrm{Peel}(v),d_\mathrm{Peel}(v')\}$, where brightness increases with higher values. Qualitatively, this picture corresponds to the edge count per unit square of the graph on the left, viewed at a larger scale.
  • Figure 4: Illustration of the door construction for a fixed realization $\omega\in\Omega_0$ where the doors $d, d', d"\in\mathcal{D}$ satisfy $d'\in(d, d")_\Gamma$. Bold black lines indicate $\mathrm{Pend}(2\ell z+[-k, k]^2)$ for $z\in\{d, d', d"\}$. The dotted black line represents a two-ended component of $G$. Informally, the red, green, and blue lines delimit the corridor, while the corresponding dotted segments serve as the doors. That $d'$ lies between $d$ and $d"$ can be seen in the picture from the fact that the red dotted segment representing $d'$ lies in the interior of the corridor section delimited by $\gamma_d$ and $\gamma_{d"}$, hatched in gray.

Theorems & Definitions (62)

  • Remark 1.1
  • Theorem I
  • Theorem II
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2: Burton-Keane argument
  • proof
  • Proposition 2.3
  • proof
  • ...and 52 more