{
  "id": "2104.01211",
  "version": "v1",
  "published": "2021-04-02T19:18:59.000Z",
  "updated": "2021-04-02T19:18:59.000Z",
  "title": "Convergence of limit shapes for 2D near-critical first-passage percolation",
  "authors": [
    "Chang-Long Yao"
  ],
  "comment": "60 pages,9 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. For each $p\\in(0,p_c)$, let $\\mathcal {B}(p)$ be the limit shape in the classical \"shape theorem\", and let $L(p)$ be the correlation length. We show that as $p\\uparrow p_c$, the rescaled limit shape $L(p)^{-1}\\mathcal {B}(p)$ converges to a Euclidean disk. This improves a result of Chayes et al. [J. Stat. Phys. 45 (1986) 933--951]. The proof relies on the scaling limit of near-critical percolation established by Garban et al. [J. Eur. Math. Soc. 20 (2018) 1195--1268], and uses the construction of the collection of continuum clusters in the scaling limit introduced by Camia et al. [Springer Proceedings in Mathematics \\& Statistics, 299 (2019) 44--89].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-02T19:18:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "2d near-critical first-passage percolation",
      "convergence",
      "bernoulli first-passage percolation",
      "scaling limit",
      "shape theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}