{
  "id": "2106.11932",
  "version": "v1",
  "published": "2021-06-22T17:12:25.000Z",
  "updated": "2021-06-22T17:12:25.000Z",
  "title": "Large deviations in random latin squares",
  "authors": [
    "Matthew Kwan",
    "Ashwin Sah",
    "Mehtaab Sawhney"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "In this note, we study large deviations of the number $\\mathbf{N}$ of $\\textit{intercalates}$ ($2\\times2$ combinatorial subsquares which are themselves Latin squares) in a random $n\\times n$ Latin square. In particular, for constant $\\delta>0$ we prove that $\\Pr(\\mathbf{N}\\le(1-\\delta)n^{2}/4)\\le\\exp(-\\Omega(n^{2}))$ and $\\Pr(\\mathbf{N}\\ge(1+\\delta)n^{2}/4)\\le\\exp(-\\Omega(n^{4/3}(\\log n)^{2/3}))$, both of which are sharp up to logarithmic factors in their exponents. As a consequence, we deduce that a typical order-$n$ Latin square has $(1+o(1))n^{2}/4$ intercalates, matching a lower bound due to Kwan and Sudakov and resolving an old conjecture of McKay and Wanless.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-22T17:12:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random latin squares",
      "study large deviations",
      "combinatorial subsquares",
      "logarithmic factors",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}