{
  "id": "cond-mat/0609210",
  "version": "v1",
  "published": "2006-09-08T19:43:58.000Z",
  "updated": "2006-09-08T19:43:58.000Z",
  "title": "Analytical calculation of neighborhood order probabilities for high dimensional Poissonic processes and mean field models",
  "authors": [
    "Cesar Augusto Sangaletti Tercariol",
    "Felipe de Mouta Kiipper",
    "Alexandre Souto Martinez"
  ],
  "comment": "12 pages, 3 figures and 1 table",
  "journal": "J. Phys. A: Math. Theor. 40 1981--1989 (2007)",
  "doi": "10.1088/1751-8113/40/9/005",
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech"
  ],
  "abstract": "Consider that the coordinates of $N$ points are randomly generated along the edges of a $d$-dimensional hypercube (random point problem). The probability that an arbitrary point is the $m$th nearest neighbor to its own $n$th nearest neighbor (Cox probabilities) plays an important role in spatial statistics. Also, it has been useful in the description of physical processes in disordered media. Here we propose a simpler derivation of Cox probabilities, where we stress the role played by the system dimensionality $d$. In the limit $d \\to \\infty$, the distances between pair of points become indenpendent (random link model) and closed analytical forms for the neighborhood probabilities are obtained both for the thermodynamic limit and finite-size system. Breaking the distance symmetry constraint drives us to the random map model, for which the Cox probabilities are obtained for two cases: whether a point is its own nearest neighbor or not.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-08T19:43:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "high dimensional poissonic processes",
      "neighborhood order probabilities",
      "mean field models",
      "probability",
      "analytical calculation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}