{
  "id": "1201.4552",
  "version": "v3",
  "published": "2012-01-22T11:49:26.000Z",
  "updated": "2012-02-07T12:40:34.000Z",
  "title": "Existence of an intermediate phase for oriented percolation",
  "authors": [
    "Hubert Lacoin"
  ],
  "comment": "16 pages, 2 figures, further typos corrected, enlarged intro and bibliography",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the following oriented percolation model of $\\mathbb {N} \\times \\mathbb{Z}^d$: we equip $\\mathbb {N}\\times \\mathbb{Z}^d$ with the edge set $\\{[(n,x),(n+1,y)] | n\\in \\mathbb {N}, x,y\\in \\mathbb{Z}^d\\}$, and we say that each edge is open with probability $p f(y-x)$ where $f(y-x)$ is a fixed non-negative compactly supported function on $\\mathbb{Z}^d$ with $\\sum_{z\\in \\mathbb{Z}^d} f(z)=1$ and $p\\in [0,\\inf f^{-1}]$ is the percolation parameter. Let $p_c$ denote the percolation threshold ans $Z_N$ the number of open oriented-paths of length $N$ starting from the origin, and study the growth of $Z_N$ when percolation occurs. We prove that for if $d\\ge 5$ and the function $f$ is sufficiently spread-out, then there exists a second threshold $p_c^{(2)}>p_c$ such that $Z_N/p^N$ decays exponentially fast for $p\\in(p_c,p_c^{(2)})$ and does not so when $p> p_c^{(2)}$. The result should extend to the nearest neighbor-model for high-dimension, and for the spread-out model when $d=3,4$. It is known that this phenomenon does not occur in dimension 1 and 2.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-02-07T12:40:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82D60",
      "60K37",
      "82B44"
    ],
    "keywords": [
      "intermediate phase",
      "non-negative compactly supported function",
      "percolation threshold ans",
      "edge set",
      "spread-out model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.4552L"
    }
  }
}