Non-self-touching paths in plane graphs

TL;DR

This paper characterizes when doubly infinite non-self-touching paths exist in the matching graph of infinite plane graphs, impacting understanding of percolation thresholds and cluster uniqueness in such graphs.

Contribution

It provides a necessary and sufficient condition for the existence of certain paths in the matching graph, advancing the theory of site percolation on infinite plane graphs.

Findings

Characterizes conditions for doubly infinite non-self-touching paths in matching graphs.

Establishes strict inequality of critical points for percolation on $G$ and $G_*$.

Shows $p_u(G) + p_c(G) \,\ge\, 1$ with equality characterized by triangulation.

Abstract

A path in a graph $G$ is called non-self-touching if two vertices are neighbours in the path if and only if they are neighbours in the graph. We investigate the existence of doubly infinite non-self-touching paths in infinite plane graphs. The matching graph $G_*$ of an infinite plane graph $G$ is obtained by adding all diagonals to all faces, and it plays an important role in the theory of site percolation on $G$. The main result of this paper is a necessary and sufficient condition on $G$ for the existence of a doubly infinite non-self-touching path in $G_*$ that traverses some diagonal. This is a key step in proving, for quasi-transitive $G$, that the critical points of site percolation on $G$ and $G_*$ satisfy the strict inequality $p_c(G_*) < p_c(G)$, and it complements the earlier result of Grimmett and Li (Random Struct. Alg. 65 (2024) 832--856), proved by different methods, concerning the case of transitive graphs. Furthermore it implies, for quasi-transitive graphs, that $p_u(G) + p_c(G) \ge 1$, with equality if and only if the graph $G_\Delta$, obtained from $G$ by emptying all separating triangles, is a triangulation. Here, $p_u$ is the critical probability for the existence of a unique infinite open cluster.