{
  "id": "2412.05503",
  "version": "v1",
  "published": "2024-12-07T02:12:00.000Z",
  "updated": "2024-12-07T02:12:00.000Z",
  "title": "Critical scaling profile for trees and connected subgraphs on the complete graph",
  "authors": [
    "Yucheng Liu",
    "Gordon Slade"
  ],
  "comment": "15 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We analyse generating functions for trees and for connected subgraphs on the complete graph, and identify a single scaling profile which applies for both generating functions in a critical window. Our motivation comes from the analysis of the finite-size scaling of lattice trees and lattice animals on a high-dimensional discrete torus, for which we conjecture that the identical profile applies in dimensions $d \\ge 8$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-07T02:12:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "60K35",
      "82B41"
    ],
    "keywords": [
      "complete graph",
      "connected subgraphs",
      "critical scaling profile",
      "high-dimensional discrete torus",
      "motivation comes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}