{
  "id": "math/0607805",
  "version": "v2",
  "published": "2006-07-31T12:42:46.000Z",
  "updated": "2007-10-31T08:15:27.000Z",
  "title": "Isoperimetric inequalities and mixing time for a random walk on a random point process",
  "authors": [
    "Pietro Caputo",
    "Alessandra Faggionato"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/07-AAP442 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2007, Vol. 17, No. 5,6, 1707-1744",
  "doi": "10.1214/07-AAP442",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the random walk on a simple point process on $\\Bbb{R}^d$, $d\\geq2$, whose jump rates decay exponentially in the $\\alpha$-power of jump length. The case $\\alpha =1$ corresponds to the phonon-induced variable-range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show, for $\\alpha\\in(0,d)$, that the random walk confined to a cubic box of side $L$ has a.s. Cheeger constant of order at least $L^{-1}$ and mixing time of order $L^2$. For the Poisson point process, we prove that at $\\alpha=d$, there is a transition from diffusive to subdiffusive behavior of the mixing time.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-10-31T08:15:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60K37",
      "82C41"
    ],
    "keywords": [
      "random walk",
      "random point process",
      "mixing time",
      "isoperimetric inequalities",
      "simple point process"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7805C"
    }
  }
}