{
  "id": "1807.09260",
  "version": "v1",
  "published": "2018-07-24T17:56:36.000Z",
  "updated": "2018-07-24T17:56:36.000Z",
  "title": "Time Correlation Exponents in Last Passage Percolation",
  "authors": [
    "Riddhipratim Basu",
    "Shirshendu Ganguly"
  ],
  "comment": "25 pages, 4 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For directed last passage percolation on $\\mathbb{Z}^2$ with exponential passage times on the vertices, let $T_{n}$ denote the last passage time from $(0,0)$ to $(n,n)$. We consider asymptotic two point correlation functions of the sequence $T_{n}$. In particular we consider ${\\rm Corr}(T_{n}, T_{r})$ for $r\\le n$ where $r,n\\to \\infty$ with $r\\ll n$ or $n-r \\ll n$. We show that in the former case ${\\rm Corr}(T_{n}, T_{r})=\\Theta((\\frac{r}{n})^{1/3})$ whereas in the latter case $1-{\\rm Corr}(T_{n}, T_{r})=\\Theta ((\\frac{n-r}{n})^{2/3})$. The argument revolves around finer understanding of polymer geometry and is expected to go through for a larger class of integrable models of last passage percolation. As by-products of the proof, we also get a couple of other results of independent interest: Quantitative estimates for locally Brownian nature of pre-limits of Airy$_2$ process coming from exponential LPP, and precise variance estimates for lengths of polymers constrained to be inside thin rectangles at the transversal fluctuation scale.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-24T17:56:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "passage percolation",
      "time correlation exponents",
      "transversal fluctuation scale",
      "point correlation functions",
      "precise variance estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}