{
  "id": "2101.08113",
  "version": "v1",
  "published": "2021-01-20T13:20:59.000Z",
  "updated": "2021-01-20T13:20:59.000Z",
  "title": "A variational formula for large deviations in First-passage percolation under tail estimates",
  "authors": [
    "Clément Cosco",
    "Shuta Nakajima"
  ],
  "comment": "This preprint supersedes arXiv:1912.13212. 31 pages, 2 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider first passage percolation with identical and independent weight distributions and first passage time ${\\rm T}$. In this paper, we study the upper tail large deviations $\\mathbb{P}({\\rm T}(0,nx)>n(\\mu+\\xi))$, for $\\xi>0$ and $x\\neq 0$ with a time constant $\\mu$ and a dimension $d$, for weights that satisfy a tail assumption $ \\beta_1\\exp{(-\\alpha t^r)}\\leq \\mathbb P(\\tau_e>t)\\leq \\beta_2\\exp{(-\\alpha t^r)}.$ When $r\\leq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $\\exp{(-(2d\\xi +o(1))n)}$. When $1< r\\leq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${\\rm T}(0,nx)>n(\\mu+\\xi)$ is described by a localization of high weights around the origin. The picture changes for $r\\geq d$ where the configuration is not anymore localized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-20T13:20:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "variational formula",
      "first-passage percolation",
      "tail estimates",
      "upper tail large deviation decays",
      "well-known eden growth model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}