{
  "id": "2308.07696",
  "version": "v1",
  "published": "2023-08-15T10:54:32.000Z",
  "updated": "2023-08-15T10:54:32.000Z",
  "title": "Scaling of Components in Critical Geometric Random Graphs on 2-dim Torus",
  "authors": [
    "Vasilii Goriachkin",
    "Tatyana Turova"
  ],
  "comment": "65 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider random graphs on the set of $N^2$ vertices placed on the discrete $2$-dimensional torus. The edges between pairs of vertices are independent, and their probabilities decay with the distance $\\rho$ between these vertices as $(N\\rho)^{-1}$. This is an example of an inhomogeneous random graph which is not of rank 1. The reported previously results on the sub- and super-critical cases of this model exhibit great similarity to the classical Erd\\H{o}s-R\\'{e}nyi graphs. Here we study the critical phase. A diffusion approximation for the size of the largest connected component rescaled with $(N^2)^{-2/3}$ is derived. This completes the proof that in all regimes the model is within the same class as Erd\\H{o}s-R\\'{e}nyi graph with respect to scaling of the largest component.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-15T10:54:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60G42",
      "60G50"
    ],
    "keywords": [
      "critical geometric random graphs",
      "dimensional torus",
      "largest component",
      "largest connected component",
      "diffusion approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}