{
  "id": "0806.3910",
  "version": "v3",
  "published": "2008-06-24T14:43:49.000Z",
  "updated": "2009-11-25T14:22:45.000Z",
  "title": "What does a random contingency table look like?",
  "authors": [
    "Alexander Barvinok"
  ],
  "comment": "25 pages, proofs simplified, results strengthened",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Let R=(r_1, ..., r_m) and C=(c_1, ..., c_n) be positive integer vectors such that r_1 +... + r_m=c_1 +... + c_n. We consider the set Sigma(R, C) of non-negative mxn integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D in Sigma(R,C) is close with high probability to a particular matrix (\"typical table'') Z defined as follows. We let g(x)=(x+1) ln(x+1)-x ln x for non-negative x and let g(X)=sum_ij g(x_ij) for a non-negative matrix X=(x_ij). Then g(X) is strictly concave and attains its maximum on the polytope of non-negative mxn matrices X with row sums R and column sums C at a unique point, which we call the typical table Z.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-11-25T14:22:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "05A16",
      "60C05",
      "15A36"
    ],
    "keywords": [
      "random contingency table look",
      "column sums",
      "row sums",
      "non-negative mxn integer matrices",
      "finite probability space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.3910B"
    }
  }
}