{
  "id": "math/0609469",
  "version": "v2",
  "published": "2006-09-16T23:09:56.000Z",
  "updated": "2007-09-06T12:03:43.000Z",
  "title": "Escape of mass in zero-range processes with random rates",
  "authors": [
    "Pablo A. Ferrari",
    "Valentin V. Sisko"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/074921707000000300 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "IMS Lecture Notes Monograph Series 2007, Vol. 55, 108-120",
  "doi": "10.1214/074921707000000300",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider zero-range processes in ${\\mathbb{Z}}^d$ with site dependent jump rates. The rate for a particle jump from site $x$ to $y$ in ${\\mathbb{Z}}^d$ is given by $\\lambda_xg(k)p(y-x)$, where $p(\\cdot)$ is a probability in ${\\mathbb{Z}}^d$, $g(k)$ is a bounded nondecreasing function of the number $k$ of particles in $x$ and $\\lambda =\\{\\lambda_x\\}$ is a collection of i.i.d. random variables with values in $(c,1]$, for some $c>0$. For almost every realization of the environment $\\lambda$ the zero-range process has product invariant measures $\\{{\\nu_{\\lambda, v}}:0\\le v\\le c\\}$ parametrized by $v$, the average total jump rate from any given site. The density of a measure, defined by the asymptotic average number of particles per site, is an increasing function of $v$. There exists a product invariant measure ${\\nu _{\\lambda, c}}$, with maximal density. Let $\\mu$ be a probability measure concentrating mass on configurations whose number of particles at site $x$ grows less than exponentially with $\\|x\\|$. Denoting by $S_{\\lambda}(t)$ the semigroup of the process, we prove that all weak limits of $\\{\\mu S_{\\lambda}(t),t\\ge 0\\}$ as $t\\to \\infty$ are dominated, in the natural partial order, by ${\\nu_{\\lambda, c}}$. In particular, if $\\mu$ dominates ${\\nu _{\\lambda, c}}$, then $\\mu S_{\\lambda}(t)$ converges to ${\\nu_{\\lambda, c}}$. The result is particularly striking when the maximal density is finite and the initial measure has a density above the maximal.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-09-06T12:03:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82C22"
    ],
    "keywords": [
      "zero-range processes",
      "random rates",
      "product invariant measure",
      "site dependent jump rates",
      "maximal density"
    ],
    "tags": [
      "monograph",
      "journal article",
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9469F"
    }
  }
}