{
  "id": "2408.04597",
  "version": "v1",
  "published": "2024-08-08T17:14:46.000Z",
  "updated": "2024-08-08T17:14:46.000Z",
  "title": "Components, large and small, are as they should be I: supercritical percolation on regular graphs of growing degree",
  "authors": [
    "Sahar Diskin",
    "Michael Krivelevich"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\\cdot d\\approx 1$. These conditions capture several well-studied graphs, such as (percolation on) the complete graph $K_n$, the binary hypercube $Q^d$, $d$-regular expanders, and random $d$-regular graphs. In particular, this serves as a unified proof for these (and other) cases. Suppose that $G$ is a $d$-regular graph on $n$ vertices, with $d=\\omega(1)$. Let $\\epsilon>0$ be a small constant, and let $p=\\frac{1+\\epsilon}{d}$. Let $y(\\epsilon)$ be the survival probability of a Galton-Watson tree with offspring distribution Po$(1+\\epsilon)$. We show that if $G$ satisfies a (very) mild edge expansion requirement, and if one has fairly good control on the expansion of small sets in $G$, then typically the percolated random subgraph $G_p$ contains a unique giant component of asymptotic order $y(\\epsilon)n$, and all the other components in $G_p$ are of order $O(\\log n/\\epsilon^2)$. We also show that this result is tight, in the sense that if one asks for a slightly weaker control on the expansion of small sets in $G$, then there are $d$-regular graphs $G$ on $n$ vertices, where typically the second largest component is of order $\\Omega(d\\log (n/d))=\\omega(\\log n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-08T17:14:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular graph",
      "growing degree",
      "supercritical percolation",
      "random subgraph",
      "small sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}