{
  "id": "1707.09628",
  "version": "v1",
  "published": "2017-07-30T15:11:02.000Z",
  "updated": "2017-07-30T15:11:02.000Z",
  "title": "A shape theorem for the scaling limit of the IPDSAW at criticality",
  "authors": [
    "Philippe Carmona",
    "Nicolas Pétrélis"
  ],
  "comment": "64 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we give a complete characterization of the scaling limit of the critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introduced in Zwanzig and Lauritzen (1968). As the system size $L$ diverges, we prove that the set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by $L^{1/3}$ converges in law for the Hausdorff distance towards a non trivial random set. This limiting set is built with a Brownian motion $B$ conditioned to come back at the origin at $a_1$ the time at which its geometric area reaches $1$. The modulus of $B$ up to $a_1$ gives the height of the limiting set, while its center of mass process is an independent Brownian motion. Obtaining the shape theorem requires to derive a functional central limit theorem for the excursion of a random walk with Laplace symmetric increments conditioned on sweeping a prescribed geometric area. This result complements the work of Denisov, Kolb and Wachtel (2015) where local central limit theorems are provided for the geometric area of the excursion of a symmetric random walk with finite second moments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-30T15:11:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B26",
      "82B41"
    ],
    "keywords": [
      "shape theorem",
      "scaling limit",
      "partially directed self-avoiding walk",
      "criticality",
      "local central limit theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 64,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}