Duality between star and plus connected components in percolation

TL;DR

This paper explores the duality between star and plus connected components in a percolation model on , showing how cycles of vacant squares relate to the finite connected components containing the origin.

Contribution

It establishes a duality relationship between star and plus connected components using the structure of outermost boundaries, extending previous work with a new geometric perspective.

Findings

Existence of plus connected cycle of vacant squares surrounding star component

Existence of star connected cycle of vacant squares surrounding plus component

Duality relationship 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. Star and plus connected components containing the origin have been previously studied using unicoherence and interface graphs. 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. There is a star connected cycle of vacant squares attached to and surrounding the finite plus connected component containing the origin.