{
  "id": "1111.3311",
  "version": "v4",
  "published": "2011-11-14T18:24:21.000Z",
  "updated": "2014-05-02T15:25:00.000Z",
  "title": "Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts",
  "authors": [
    "Leonid V. Bogachev"
  ],
  "comment": "Minor editorial corrections. Published in \"Random Structures and Algorithms\" (18 Apr 2014, Early View, online), http://onlinelibrary.wiley.com/doi/10.1002/rsa.20540/abstract",
  "doi": "10.1002/rsa.20540",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form $\\mathcal{F}(z)=\\prod_{\\ell=1}^\\infty \\mathcal{F}_0(z^\\ell)$ (which entails equal weighting among possible parts $\\ell\\in\\mathbb{N}$). Under mild technical assumptions on the function $H_0(u)=\\ln(\\mathcal{F}_0(u))$, we show that the limit shape $\\omega^*(x)$ exists and is given by the equation $y=\\gamma^{-1}H_0(\\mathrm{e}^{-\\gamma x})$, where $\\gamma^2=\\int_0^1 u^{-1}H_0(u)\\,\\mathrm{d}u$. The wide class of partition measures covered by this result includes (but is not limited to) representatives of the three meta-types of decomposable combinatorial structures --- assemblies, multisets and selections. Our method is based on the usual randomization and conditioning; to this end, a suitable local limit theorem is proved. The proofs are greatly facilitated by working with the cumulants of sums of the part counts rather than with their moments.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-05-02T15:25:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "60C05",
      "60F05",
      "60G50"
    ],
    "keywords": [
      "limit shape",
      "random integer partitions",
      "multiplicative ensembles",
      "unified derivation",
      "equiweighted parts"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.3311B"
    }
  }
}