Outermost boundaries for star-connected components in percolation

TL;DR

This paper characterizes the outermost boundary of finite star-connected components in percolation on a grid, showing it forms a union of disjoint cycles and providing an inductive method to construct a containing circuit.

Contribution

It proves the structure of the outermost boundary for star-connected components and introduces an inductive procedure to form a circuit encompassing all boundary edges.

Findings

The outermost boundary is a union of cycles with disjoint interiors.

Each pair of cycles shares at most one vertex.

An inductive method to construct a circuit containing all boundary edges.

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. If the occupied plus-connected component \(C^+(0)\) containing the origin is finite, it is known that the outermost boundary \(\partial^+_0\) of \(C^+(0)\) is a unique cycle surrounding the origin. For the finite occupied star-connected component \(C(0)\) containing the origin, we prove in this paper that the outermost boundary \(\partial_0\) is a unique connected graph consisting of a union of cycles \(\cup_{1 \leq i \leq n} C_i\) with mutually disjoint interiors. Moreover, we have that each pair of cycles in \(\partial_0\) share at most one vertex in common and we provide an inductive procedure to obtain a circuit containing all the edges of \(\cup_{1 \leq i \leq n} C_i.\) This has applications for contour analysis of star-connected components in percolation.