# Duality in percolation via outermost boundaries III: Plus connected components

## TL;DR

This paper explores the duality between star and plus connected components in a percolation model on a grid, revealing how vacant squares form cycles around finite occupied clusters using boundary structures.

## Contribution

It introduces a duality framework between star and plus connected components in percolation, utilizing outer boundary structures to characterize their relationship.

## Key findings

- Existence of star connected cycles surrounding finite plus connected components.
- Duality between star and plus connected components in the percolation model.
- Application of boundary structures to analyze connectivity properties.

## 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.