{
  "id": "2302.00031",
  "version": "v1",
  "published": "2023-01-31T19:03:49.000Z",
  "updated": "2023-01-31T19:03:49.000Z",
  "title": "A new proof of the bunkbed conjecture in the $p\\uparrow 1$ limit",
  "authors": [
    "Lawrence Hollom"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "For a finite simple graph $G$, the bunkbed graph $G^\\pm$ is defined to be the product graph $G\\square K_2$. We will label the two copies of a vertex $v\\in V(G)$ as $v_-$ and $v_+$. The bunkbed conjecture, posed by Kasteleyn, states that for independent bond percolation on $G^\\pm$, percolation from $u_-$ to $v_-$ is at least as likely as percolation from $u_-$ to $v_+$, for any $u,v\\in V(G)$. Despite the plausibility of this conjecture, so far the problem in full generality remains open. Recently, Hutchcroft, Nizi\\'{c}-Nikolac, and Kent gave a proof of the conjecture in the $p\\uparrow 1$ limit. Here we present a new proof of the bunkbed conjecture in this limit, working in the more general setting of allowing different probabilities on different edges of $G^\\pm$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-31T19:03:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80"
    ],
    "keywords": [
      "bunkbed conjecture",
      "full generality remains open",
      "independent bond percolation",
      "finite simple graph",
      "product graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}