{
  "id": "2308.10267",
  "version": "v1",
  "published": "2023-08-20T13:44:00.000Z",
  "updated": "2023-08-20T13:44:00.000Z",
  "title": "Percolation through Isoperimetry",
  "authors": [
    "Sahar Diskin",
    "Joshua Erde",
    "Mihyun Kang",
    "Michael Krivelevich"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We determine necessary and sufficient conditions on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\\frac{1}{d}$ which resembles the emergence of a giant component in the binomial random graph model $G(n,p)$. More precisely, let $d=\\omega(1)$, let $\\epsilon>0$ be a small enough constant, and let $p \\cdot d=1+\\epsilon$. We show that if $C$ is sufficiently large and $G$ is a $d$-regular $n$-vertex graph where every subset $S\\subseteq V(G)$ of order at most $\\frac{n}{2}$ has edge-boundary of size at least $C|S|$, then $G_p$ typically has a unique component of linear order, whose order is asymptotically $y(\\epsilon)n$, where $y(\\epsilon)$ is the survival probability of a Galton-Watson tree with offspring distribution Po$(1+\\epsilon)$. We further give examples to show that this result is tight both in terms of its dependence on $C$, and with respect to the order of the second-largest component, which is in contrast to the case of constant $d$. We also consider a more general setting, where we only control the expansion of sets up to size $k$. In this case, we show that if $G$ is such that every subset $S\\subseteq V(G)$ of order at most $k$ has edge-boundary of size at least $d|S|$ and $p$ is such that $p\\cdot d \\geq 1 + \\epsilon$, then $G_p$ typically contains a component of order $\\Omega(k)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-20T13:44:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60K35",
      "82B43"
    ],
    "keywords": [
      "percolation",
      "isoperimetry",
      "binomial random graph model",
      "vertex graph",
      "random subgraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}