Duality in percolation via outermost boundaries II: Star connected components and left right crossings

TL;DR

This paper explores duality relationships between star and plus connected components in a percolation model on a grid, establishing the existence of dual cycles and crossing properties in rectangles.

Contribution

It introduces a duality framework between star and plus connected components using outermost boundaries, extending previous work to include new crossing and boundary properties.

Findings

Existence of plus connected cycles surrounding star connected components.

Mutual exclusivity of left-right and top-down crossings in rectangles.

Duality between star and plus connected components in percolation.

Abstract

Tile \(\mathbb{R}^2\) into disjoint unit squares \(\{S_k\}_{k \geq 0}\) with the origin being the centre of \(S_0\) and say that \(S_i\) and \(S_j\) are star adjacent if they share a corner and plus adjacent if they share an edge. Every square is either vacant or occupied. In this paper, we use the structure of the outermost boundaries derived in Ganesan (2015) to alternately obtain duality between star and plus connected components in the following sense: There is a plus connected cycle of vacant squares attached to surrounding the finite star connected component containing the origin. We also obtain the mutual exclusivity of left right crossings and top down crossings of star and plus connected components in rectangles.