{
  "id": "1904.12009",
  "version": "v1",
  "published": "2019-04-26T18:26:14.000Z",
  "updated": "2019-04-26T18:26:14.000Z",
  "title": "Universality of the time constant for $2D$ critical first-passage percolation",
  "authors": [
    "Michael Damron",
    "Jack Hanson",
    "Wai-Kit Lam"
  ],
  "comment": "29 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider first-passage percolation (FPP) on the triangular lattice with vertex weights $(t_v)$ whose common distribution function $F$ satisfies $F(0)=1/2$. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by $T(0,\\partial B(n))$ the first-passage time from $0$ to $\\{x : \\|x\\|_\\infty = n\\}$, we show existence of the \"time constant'' and find its exact value to be \\[ \\lim_{n \\to \\infty} \\frac{T(0,\\partial B(n))}{\\log n} = \\frac{I}{2\\sqrt{3}\\pi} \\text{ almost surely}, \\] where $I = \\inf\\{x > 0 : F(x) > 1/2\\}$ and $F$ is any critical distribution for $t_v$. This result shows that the time constant is universal and depends only on the value of $I$. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of $I$, under the optimal moment condition on $F$. The proof method also shows an analogous universality on other two-dimensional lattices, assuming the time constant exists.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-26T18:26:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "time constant",
      "critical first-passage percolation",
      "universality",
      "exact value",
      "optimal moment condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}