{
  "id": "1508.06443",
  "version": "v1",
  "published": "2015-08-26T11:15:40.000Z",
  "updated": "2015-08-26T11:15:40.000Z",
  "title": "Outermost boundaries for star-connected components in percolation",
  "authors": [
    "Ghurumuruhan Ganesan"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-26T11:15:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "outermost boundary",
      "percolation",
      "disjoint unit squares",
      "contour analysis",
      "finite occupied star-connected component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150806443G"
    }
  }
}