{
  "id": "1003.1289",
  "version": "v2",
  "published": "2010-03-05T15:22:20.000Z",
  "updated": "2010-12-07T06:57:21.000Z",
  "title": "On the critical parameter of interlacement percolation in high dimension",
  "authors": [
    "Alain-Sol Sznitman"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/10-AOP545 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2011, Vol. 39, No. 1, 70-103",
  "doi": "10.1214/10-AOP545",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The vacant set of random interlacements on ${\\mathbb{Z}}^d$, $d\\ge3$, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039--2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831--858] that there is a nondegenerate critical value $u_*$ such that the vacant set at level $u$ percolates when $u<u_*$ and does not percolate when $u>u_*$. We derive here an asymptotic upper bound on $u_*$, as $d$ goes to infinity, which complements the lower bound from Sznitman [Probab. Theory Related Fields, to appear]. Our main result shows that $u_*$ is equivalent to $\\log d$ for large $d$ and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on $2d$-regular trees, which has been explicitly computed in Teixeira [Electron. J. Probab. 14 (2009) 1604--1627].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-12-07T06:57:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical parameter",
      "high dimension",
      "interlacement percolation",
      "vacant set",
      "random interlacements"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.1289S"
    }
  }
}