{
  "id": "1402.4983",
  "version": "v3",
  "published": "2014-02-20T12:29:38.000Z",
  "updated": "2015-03-13T11:24:26.000Z",
  "title": "Two-site localisation in the Bouchaud trap model with slowly varying traps",
  "authors": [
    "Stephen Muirhead"
  ],
  "comment": "13 pages",
  "journal": "Electron. Commun. Probab. 20 (2015), no. 25, 1--15",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the Bouchaud trap model on the integers in the case that the trap distribution has a slowly varying tail at infinity. We prove that the model eventually localises on exactly two sites with overwhelming probability. This is a stronger form of localisation than has previously been established in the literature for the Bouchaud trap model on the integers in the case of regularly varying traps. Underlying this result is the fact that the sum of a sequence of i.i.d. random variables with a slowly varying tail is asymptotically dominated by the maximal term.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-04T08:17:27.000Z",
      "title": "Two-site localisation in the Bouchaud trap model with slowly-varying traps",
      "abstract": "We consider the Bouchaud trap model on the integers in the case where the trapping landscape consists of independent identically distributed random variables with a slowly-varying tail, that is, whose survival function has a reciprocal that is slowly-varying at infinity. Under these conditions, we prove that the model eventually localises on exactly two sites with overwhelming probability. This is a stronger form of localisation than has previously been established in the literature for the Bouchaud trap model on the integers in the case of heavy-tailed traps. The fact underlying the result is that the running maximum of a sequence of i.i.d. random variables with a slowly-varying tail asymptotically dominates the cumulative sum of the sequence, with overwhelming probability. We further prove that the proportion of probability mass on each of the two localisation sites is uniformly distributed in the limit, as well as deriving a limit formula for the distance of the localisation sites from the origin.",
      "comment": "13 pages. Substantial revision, removing various technical assumptions",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-03-13T11:24:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bouchaud trap model",
      "two-site localisation",
      "slowly-varying traps",
      "localisation sites",
      "independent identically distributed random variables"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.4983M"
    }
  }
}