{
  "id": "1712.04548",
  "version": "v1",
  "published": "2017-12-12T22:10:08.000Z",
  "updated": "2017-12-12T22:10:08.000Z",
  "title": "Accessible Percolation with Crossing Valleys on $n$-ary Trees",
  "authors": [
    "Frank Duque",
    "Alejandro Roldán-Correa",
    "Leon A. Valencia"
  ],
  "comment": "12 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, also motivated by evolutionary biology and evolutionary computation, we study a variation of the accessibility percolation model. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leave, where any $k$ consecutive vertices in the path contain at least one vertex of the subsequence. An $n$-ary tree, with height $h$, is a tree whose vertices at distance at most $h-1$ to the root have $n$ children. For the case of $n$-ary trees, we proof that, as $h$ tends to infinity the probability of having such subsequence: tends to 1, if $n(h)\\geq c\\sqrt[k]{h/(ek)} $ and $c>1$; and tends to 0, if $n(h)\\leq c\\sqrt[k]{h/(ek)} $ and $c<1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-12T22:10:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60C05",
      "92D15"
    ],
    "keywords": [
      "ary tree",
      "accessible percolation",
      "crossing valleys",
      "accessibility percolation model",
      "evolutionary biology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}