Outermost boundaries for star-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. 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.