{
  "id": "1211.0504",
  "version": "v4",
  "published": "2012-11-02T17:56:41.000Z",
  "updated": "2015-05-14T06:38:32.000Z",
  "title": "Stein's method and the rank distribution of random matrices over finite fields",
  "authors": [
    "Jason Fulman",
    "Larry Goldstein"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/13-AOP889 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2015, Vol. 43, No. 3, 1274-1314",
  "doi": "10.1214/13-AOP889",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "With ${\\mathcal{Q}}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\\times(n+m)$ matrices over the finite field $\\mathbb{F}_q$ of size $q\\ge2$, and ${\\mathcal{Q}}_q$ the distributional limit of ${\\mathcal{Q}}_{q,n}$ as $n\\rightarrow\\infty$, we apply Stein's method to prove the total variation bound $\\frac{1}{8q^{n+m+1}}\\leq\\|{\\mathcal{Q}}_{q,n}-{\\mathcal{Q}}_q\\|_{\\mathrm{TV}}\\leq\\frac{3}{q^{n+m+1}}$. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-05T03:20:22.000Z",
      "abstract": "With Q_{q,n} the distribution of n minus the rank of a matrix chosen uniformly from the collection of all n x (n+m) matrices over the finite field F_q of size q at least 2, and Q_q the distributional limit of Q_{q,n} as n tends to infinity, we apply Stein's method to prove the total variation bound 1/(8q^{n+m+1}) \\le ||Q_{q,n}-Q_q||_{TV} \\le 3/q^{n+m+1}. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric, and Hermitian matrices.",
      "comment": "Version 2 includes five the new examples of the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric, and Hermitian matrices, increasing the length of the paper from 12 to 37 pages. Version 3 includes additional clarifications and generalizes the uniform distribution over matrices with entries in F_q from the square to the rectangular case",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-05-14T06:38:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "steins method",
      "rank distribution",
      "finite field",
      "random matrices",
      "similar sharp results"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.0504F"
    }
  }
}