Connected components and topological ends of stationary planar forests

TL;DR

This paper classifies the topological structures of stationary planar forests, showing that trees have at most two ends and providing explicit examples for all possible configurations, with applications to various models.

Contribution

It offers a complete classification of the topological structures of stationary planar forests, including explicit constructions for all admissible configurations.

Findings

All trees have at most two topological ends.

The number of one-ended and two-ended components is constrained by the model.

Explicit examples realize all compatible topological structures.

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. The latter extends a theorem of Chaika and Krishnan (2019), who considered a lattice setting. Our proofs combine classical Burton-Keane type arguments with substantial new conceptual ideas using planar topology, resulting in a robust, unified approach.