{
  "id": "1004.1251",
  "version": "v1",
  "published": "2010-04-08T06:28:18.000Z",
  "updated": "2010-04-08T06:28:18.000Z",
  "title": "Long-range percolation on the hierarchical lattice",
  "authors": [
    "Vyacheslav Koval",
    "Ronald Meester",
    "Pieter Trapman"
  ],
  "comment": "24 pages",
  "journal": "Electronic Journal of Probability [Online], 17 (2012): 1-21",
  "doi": "10.1214/EJP.v17-1977",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\\exp(-\\beta^{-k} \\alpha)$, independently of all other edges. For fixed $\\beta$, we show that the critical value $\\alpha_c(\\beta)$ is non-trivial if and only if $N < \\beta < N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $\\alpha_c(\\beta)$ as a function of $\\beta$. This means that the phase diagram of this model is well understood.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-08T06:28:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B28"
    ],
    "keywords": [
      "hierarchical lattice",
      "study long-range percolation",
      "infinite component",
      "percolation probability",
      "phase diagram"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.1251K"
    }
  }
}