{
  "id": "1408.3779",
  "version": "v2",
  "published": "2014-08-17T00:38:21.000Z",
  "updated": "2014-08-23T23:05:06.000Z",
  "title": "Painlevé representation of Tracy-Widom$_β$ distribution for $β = 6$",
  "authors": [
    "Igor Rumanov"
  ],
  "comment": "mistakes corrected, references added",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "math.PR",
    "nlin.SI"
  ],
  "abstract": "In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of $\\beta$. Using this general result, the case $\\beta=6$ is further considered here. This is the smallest even $\\beta$, when the corresponding Lax pair and its relation to Painlev\\'e II (PII) have not been known before, unlike cases $\\beta=2$ and $4$. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ODE for the logarithmic derivative of Tracy-Widom distribution for $\\beta=6$ involving the PII function in the coefficients, is found, which allows one to compute asymptotics for the distribution function. The ODE is a consequence of a linear system of three ODEs for which the local Painlev\\'e analysis yields series solutions with exponents in the set $4/3$, $1/3$ and $-2/3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-17T00:38:21.000Z",
      "abstract": "In \\cite{betaFP1}, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of $\\beta$. Using this general result, the case $\\beta=6$ is further considered here. This is the smallest even $\\beta$, when the corresponding Lax pair and its relation to Painlev\\'e II (PII) have not been known before, unlike cases $\\beta=2$ and $4$. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ODE for the logarithmic derivative of Tracy-Widom distribution for $\\beta=6$ involving the PII function in the coefficients, is found, which allows one to compute asymptotics for the distribution function. The ODE is a consequence of a linear system of three ODEs for which the local Painlev\\'e analysis yields series solutions with exponents in the set $4/3$, $1/3$ and $-2/3$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-23T23:05:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distribution",
      "painleve analysis yields series solutions",
      "tracy-widom",
      "local painleve analysis yields series",
      "second order nonlinear ode"
    ],
    "publication": {
      "doi": "10.1007/s00220-015-2487-5",
      "journal": "Communications in Mathematical Physics",
      "year": 2016,
      "month": "Mar",
      "volume": 342,
      "number": 3,
      "pages": 843
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1311352,
      "adsabs": "2016CMaPh.342..843R"
    }
  }
}