{
  "id": "cond-mat/0509187",
  "version": "v2",
  "published": "2005-09-07T21:18:38.000Z",
  "updated": "2006-01-04T18:00:33.000Z",
  "title": "Stochastic Loewner evolution driven by Levy processes",
  "authors": [
    "I. Rushkin",
    "P. Oikonomou",
    "L. P. Kadanoff",
    "I. A. Gruzberg"
  ],
  "comment": "Published version, minor typos corrected, added references",
  "journal": "J. Stat. Mech. (2006) P01001",
  "doi": "10.1088/1742-5468/2006/01/P01001",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches. We consider a generalized SLE driven by a superposition of a Brownian motion and a stable Levy process. The situation is defined by the usual SLE parameter, $\\kappa$, as well as $\\alpha$ which defines the shape of the stable Levy distribution. The resulting behavior is characterized by two descriptors: $p$, the probability that the trace self-intersects, and $\\tilde{p}$, the probability that it will approach arbitrarily close to doing so. Using Dynkin's formula, these descriptors are shown to change qualitatively and singularly at critical values of $\\kappa$ and $\\alpha$. It is reasonable to call such changes ``phase transitions''. These transitions occur as $\\kappa$ passes through four (a well-known result) and as $\\alpha$ passes through one (a new result). Numerical simulations are then used to explore the associated touching and near-touching events.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-01-04T18:00:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic loewner evolution driven",
      "levy processes",
      "standard stochastic loewner evolution",
      "brownian motion",
      "usual sle parameter"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2006,
      "month": "Jan",
      "volume": 2006,
      "number": 1,
      "pages": 1001
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006JSMTE..01..001R"
    }
  }
}