{
  "id": "math/0702053",
  "version": "v2",
  "published": "2007-02-02T14:12:24.000Z",
  "updated": "2007-02-05T10:03:46.000Z",
  "title": "Large deviations for empirical path measures in cycles of integer partitions",
  "authors": [
    "Stefan Adams"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a large system of $N$ Brownian motions in $\\mathbb{R}^d$ on some fixed time interval $[0,\\beta]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $\\sigma(i)$-th motion, where $\\sigma$ is a uniformly distributed random permutation of $1,...,N$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) when $ \\Lambda\\uparrow\\mathbb{R}^d $ and $ N/|\\Lambda|\\to\\rho $. The rate function is given as a variational formula involving a certain entropy functional and a Fenchel-Legendre transform. Depending on the dimension and the density $ \\rho $, there is phase transition behaviour for the empirical path measure. For certain parameters (high density, large time horizon) and dimensions $ d\\ge 3 $ the empirical path measure is not supported on all paths $ [0,\\infty)\\to\\mathbb{R}^d $ which contain a bridge path of any finite multiple of the time horizon $ [0,\\beta] $. For dimensions $ d=1,2 $, and for small densities and small time horizon $ [0,\\beta] $ in dimensions $ d\\ge 3$, the empirical path measure is supported on those paths. In the first regime a finite fraction of the motions lives in cycles of infinite length. We outline that this transition leads to an empirical path measure interpretation of {\\it Bose-Einstein condensation}, known for systems of Bosons.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-02-05T10:03:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60J65",
      "82B10",
      "82B26"
    ],
    "keywords": [
      "large deviations",
      "integer partitions",
      "th motion",
      "large time horizon",
      "small time horizon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......2053A"
    }
  }
}