Duality in percolation via outermost boundaries III: Plus connected components

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 (2017) to alternately obtain duality between star and plus connected components in the following sense: There is a star connected cycle of vacant squares attached to and surrounding the finite plus connected component containing the origin.