{
  "id": "cond-mat/0511296",
  "version": "v2",
  "published": "2005-11-11T19:10:25.000Z",
  "updated": "2005-11-19T17:49:32.000Z",
  "title": "Directed percolation in two dimensions: An exact solution",
  "authors": [
    "L. C. Chen",
    "F. Y. Wu"
  ],
  "comment": "Figures now included, one reference added",
  "journal": "in \"Differential Geometry and Physics\", Nankai Tracts in Math. Vol. 10, Eds. M. L. Ge and W. Zhang (World Scientific, Singapore, 2006) pp. 160-168.",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider a directed percolation process on an ${\\cal M}$ x ${\\cal N}$ rectangular lattice whose vertical edges are directed upward with an occupation probability y and horizontal edges directed toward the right with occupation probabilities x and 1 in alternate rows. We deduce a closed-form expression for the percolation probability P(x,y), the probability that one or more directed paths connect the lower-left and upper-right corner sites of the lattice. It is shown that P(x,y) is critical in the aspect ratio $a = {\\cal M}/{\\cal N}$ at a value $a_c =[1-y^2-x(1-y)^2]/2y^2$ where P(x,y) is discontinuous, and the critical exponent of the correlation length for $a < a_c$ is $\\nu=2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-11-19T17:49:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact solution",
      "dimensions",
      "occupation probability",
      "upper-right corner sites",
      "closed-form expression"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005cond.mat.11296C"
    }
  }
}