{
  "id": "1810.08270",
  "version": "v1",
  "published": "2018-10-18T20:38:23.000Z",
  "updated": "2018-10-18T20:38:23.000Z",
  "title": "Lower bounds for fluctuations in first-passage percolation for general distributions",
  "authors": [
    "Michael Damron",
    "Jack Hanson",
    "Christian Houdré",
    "Chen Xu"
  ],
  "comment": "27 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In first-passage percolation (FPP), one assigns i.i.d.~weights to the edges of the cubic lattice $\\mathbb{Z}^d$ and analyzes the induced weighted graph metric. If $T(x,y)$ is the distance between vertices $x$ and $y$, then a primary question in the model is: what is the order of the fluctuations of $T(0,x)$? It is expected that the variance of $T(0,x)$ grows like the norm of $x$ to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order $\\log \\|x\\|$. This result was found in the '90s and there has not been any improvement since. In this paper, we address the problem of getting stronger fluctuation bounds: to show that $T(0,x)$ is with high probability not contained in an interval of size $o(\\log \\|x\\|)^{1/2}$, and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of \"hi-mode\" (large).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-18T20:38:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first-passage percolation",
      "general distributions",
      "getting stronger fluctuation bounds",
      "special edge-weight distributions",
      "general edge-weight distributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}