{
  "id": "1308.2656",
  "version": "v2",
  "published": "2013-08-12T19:34:02.000Z",
  "updated": "2013-12-03T23:11:13.000Z",
  "title": "Partially observed Boolean sequences and noise sensitivity",
  "authors": [
    "Daniel Ahlberg"
  ],
  "comment": "13 pages",
  "doi": "10.1017/S0963548314000030",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{H}$ denote a collection of subsets of $\\{1,2,\\ldots,n\\}$, and assign independent random variables uniformly distributed over $[0,1]$ to the $n$ elements. Declare an element $p$-present if its corresponding value is at most $p$. In this paper, we quantify how much the observation of the $r$-present ($r>p$) set of elements affects the probability that the set of $p$-present elements is contained in $\\mathcal{H}$. In the context of percolation, we find that this question is closely linked to the near-critical regime. As a consequence, we show that for every $r>1/2$, bond percolation on the subgraph of the square lattice given by the set of $r$-present edges is almost surely noise sensitive at criticality, thus generalizing a result due to Benjamini, Kalai and Schramm.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-03T23:11:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60K35",
      "06E30"
    ],
    "keywords": [
      "noise sensitivity",
      "boolean sequences",
      "assign independent random variables"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.2656A"
    }
  }
}