{
  "id": "1312.0751",
  "version": "v3",
  "published": "2013-12-03T09:34:44.000Z",
  "updated": "2015-10-01T13:50:26.000Z",
  "title": "The quenched limiting distributions of a charged-polymer model",
  "authors": [
    "Nadine Guillotin-Plantard",
    "Renato Soares Dos Santos"
  ],
  "comment": "23 pages. v2->v3: Title corrected, some improvements for readability added",
  "categories": [
    "math.PR"
  ],
  "abstract": "The limit distributions of the charged-polymer Hamiltonian of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths and Higgs [Gaussian case] are considered. Two sources of randomness enter in the definition: a random field $q= (q_i)_{i\\geq 1}$ of i.i.d. random variables (called random charges) and a random walk $S = (S_n)_{n \\in \\mathbb{N}}$ evolving in $\\mathbb{Z}^d$, independent of the charges. The energy or Hamiltonian $K = (K_n)_{n \\geq 2}$ is then defined as $$K_n := \\sum_{1\\leq i < j\\leq n} q_i q_j {\\bf 1}_{\\{S_i=S_j\\}}.$$ The law of $K$ under the joint law of $q$ and $S$ is called \"annealed\", and the conditional law given $q$ is called \"quenched\". Recently, strong approximations under the annealed law were proved for $K$. In this paper we consider the limit distributions of $K$ under the quenched law.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-22T19:33:58.000Z",
      "title": "The quenched limiting distributions of a charged-polymer model in one and two dimensions",
      "abstract": "The limit distributions of the charged-polymer Hamiltonian of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths and Higgs [Gaussian case] are considered. Two sources of randomness enter in the definition: a random field $q= (q_i)_{i\\geq 1}$ of i.i.d.\\ random variables, which is called the random \\emph{charges}, and a random walk $S = (S_n)_{n \\in \\NN}$ evolving in $\\ZZ^d$, independent of the charges. The energy or Hamiltonian $K = (K_n)_{n \\geq 2}$ is then defined as $$K_n := \\sum_{1\\leq i < j\\leq n} q_i q_j {\\bf 1}_{\\{S_i=S_j\\}}.$$ The law of $K$ under the joint law of $q$ and $S$ is called \"annealed\", and the conditional law given $q$ is called \"quenched\". Recently, strong approximations under the annealed law were proved for $K$. In this paper we consider the limit distributions of $K$ under the quenched law.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-10-01T13:50:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quenched limiting distributions",
      "charged-polymer model",
      "limit distributions",
      "dimensions",
      "gaussian case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.0751G"
    }
  }
}