{
  "id": "2311.16631",
  "version": "v1",
  "published": "2023-11-28T09:31:44.000Z",
  "updated": "2023-11-28T09:31:44.000Z",
  "title": "Climbing up a random subgraph of the hypercube",
  "authors": [
    "Michael Anastos",
    "Sahar Diskin",
    "Dor Elboim",
    "Michael Krivelevich"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Let $Q^d$ be the $d$-dimensional binary hypercube. We say that $P=\\{v_1,\\ldots, v_k\\}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\\in [k-1]$ the edge $v_iv_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to a one coordinate in $v_{i+1}$. Form a random subgraph $Q^d_p$ by retaining each edge in $E(Q^d)$ independently with probability $p$. We show that there is a phase transition with respect to the length of a longest increasing path around $p=\\frac{e}{d}$. Let $\\alpha$ be a constant and let $p=\\frac{\\alpha}{d}$. When $\\alpha<e$, then there exists a $\\delta \\in [0,1)$ such that whp a longest increasing path in $Q^d_p$ is of length at most $\\delta d$. On the other hand, when $\\alpha>e$, whp there is a path of length $d-2$ in $Q^d_p$, and in fact, whether it is of length $d-2, d-1$, or $d$ depends on whether the all-zero and all-one vertices percolate or not.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-28T09:31:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "05C80"
    ],
    "keywords": [
      "random subgraph",
      "longest increasing path",
      "all-one vertices percolate",
      "dimensional binary hypercube",
      "zero coordinate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}