{
  "id": "math/0611242",
  "version": "v1",
  "published": "2006-11-08T20:35:11.000Z",
  "updated": "2006-11-08T20:35:11.000Z",
  "title": "Hitting time of large subsets of the hypercube",
  "authors": [
    "Jiri Cerny",
    "Veronique Gayrard"
  ],
  "comment": "14 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the simple random walk on the $n$-dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly-rescaled hitting time is asymptotically exponentially distributed, uniformly in the starting position of the walk. These conditions are then verified for percolation clouds with densities that are much smaller than $(n \\log n)^{-1}$. A main motivation behind this paper is the study of the so-called aging phenomenon in the Random Energy Model (REM), the simplest model of a mean-field spin glass. Our results allow us to prove aging in the REM for all temperatures, thereby extending earlier results to their optimal temperature domain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-11-08T20:35:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hitting time",
      "large subsets",
      "optimal temperature domain",
      "mean-field spin glass",
      "random energy model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11242C"
    }
  }
}