{
  "id": "1809.06792",
  "version": "v1",
  "published": "2018-09-18T15:20:43.000Z",
  "updated": "2018-09-18T15:20:43.000Z",
  "title": "On the combinatorics of last passage percolation in a quarter square and $\\mathrm{GOE}^2$ fluctuations",
  "authors": [
    "Dan Betea"
  ],
  "comment": "16 pages, 4 figures",
  "categories": [
    "math-ph",
    "math.CO",
    "math.MP",
    "math.PR"
  ],
  "abstract": "In this note we give a(nother) combinatorial proof of an old result of Baik--Rains: that for appropriately considered independent geometric weights, the generating series for last passage percolation polymers in a $2n \\times n \\times n$ quarter square (point-to-half-line-reflected geometry) splits as the product of two simpler generating series---that for last passage percolation polymers in a point-to-line geometry and that for last passage percolation in a point-to-point-reflected (half-space) geometry, the latter both in an $n \\times n \\times n$ triangle. As a corollary, for iid geometric random variables---of parameter $q$ off-diagonal and parameter $\\sqrt{q}$ on the diagonal---we see that the last passage percolation time in said quarter square obeys Tracy--Widom $\\mathrm{GOE}^2$ fluctuations in the large $n$ limit as both the point-to-line and the point-to-point-reflected geometries have known GOE fluctuations. This is a discrete analogue of a celebrated Baik--Rains theorem (the limit $q \\to 0$) and more recently of results from Bisi's PhD thesis (the limit $q \\to 1$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-18T15:20:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fluctuations",
      "passage percolation polymers",
      "iid geometric random variables-of parameter",
      "combinatorics",
      "quarter square obeys tracy-widom"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}