Duality and Outermost Boundaries in Generalized Percolation Lattices

TL;DR

This paper investigates the geometric and topological properties of percolation lattices on planar graphs, focusing on boundary structures, duality, and crossing probabilities in percolation models.

Contribution

It introduces conditions for cellular structures in planar graphs to exhibit lattice-like percolation behavior and explores duality and boundary cycles in this context.

Findings

Characterization of outermost boundaries of connected components.

Conditions for dual graphs to also form percolation lattices.

Analysis of crossing probabilities in rectangular regions.

Abstract

In this paper we consider a connected planar graph $G$ and impose conditions that results in $G$ having a percolation lattice-like cellular structure. Assigning each cell of $G$ to be either occupied or vacant, we describe the outermost boundaries of star and plus connected components in $G$. We then consider the dual graph of $G$ and impose conditions under which the dual is also a percolation lattice. Finally, using $G$ and its dual, we construct vacant cell cycles surrounding occupied components and study left right crossings and bond percolation in rectangles.