{
  "id": "1407.5402",
  "version": "v2",
  "published": "2014-07-21T07:48:36.000Z",
  "updated": "2014-10-06T12:10:24.000Z",
  "title": "From Sine kernel to Poisson statistics",
  "authors": [
    "Romain Allez",
    "Laure Dumaz"
  ],
  "comment": "24 pages, 5 figures",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech"
  ],
  "abstract": "We study the Sine$_\\beta$ process introduced in [B. Valk\\'o and B. Vir\\'ag. Invent. math. (2009)] when the inverse temperature $\\beta$ tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of $\\beta$-ensembles and its law is characterized in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine$_\\beta$ point process converges weakly to a Poisson point process on $\\mathbb{R}$. Thus, the Sine$_\\beta$ point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to $\\beta=\\infty$) and the Poisson process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-21T07:48:36.000Z",
      "abstract": "We study the Sine$_\\beta$ process introduced in [B. Valk\\'o and B. Vir\\'ag. Invent. math. (2009)] when the inverse temperature $\\beta$ tends to $0$. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of $\\beta$-ensembles and its law is characterized in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine$_\\beta$ point process converges weakly to a Poisson point process on $\\mathbb{R}$. Thus, the Sine$_\\beta$ point processes establish a smooth crossover between the Sine kernel process (corresponding to $\\beta=2$) and the Poisson process.",
      "comment": "22 pages, 5 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-06T12:10:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "poisson statistics",
      "poisson point process",
      "eigenvalues point process",
      "sine kernel process",
      "inverse temperature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5402A"
    }
  }
}