{
  "id": "1607.02821",
  "version": "v1",
  "published": "2016-07-11T04:46:30.000Z",
  "updated": "2016-07-11T04:46:30.000Z",
  "title": "Discontinuity in the asymptotic behavior of planar orthogonal polynomials under a perturbation of the Gaussian weight",
  "authors": [
    "Seung-Yeop Lee",
    "Meng Yang"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the orthogonal polynomials, $\\{P_n(z)\\}_{n=0,1,\\cdots}$, with respect to the measure $$|z-a|^{2c} e^{-N|z|^2}dA(z)$$ supported over the whole complex plane, where $a>0$, $N>0$ and $c>-1$. We look at the scaling limit where $n$ and $N$ tend to infinity while keeping their ratio, $n/N$, fixed. The support of the limiting zero distribution is given in terms of certain \"limiting potential-theoretic skeleton\" of the unit disk. We show that, as we vary $c$, both the skeleton and the asymptotic distribution of the zeros behave discontinuously at $c=0$. The smooth interpolation of the discontinuity is obtained by the further scaling of $c=e^{-\\eta N}$ in terms of the parameter $\\eta\\in[0,\\infty).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-11T04:46:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "planar orthogonal polynomials",
      "asymptotic behavior",
      "gaussian weight",
      "discontinuity",
      "perturbation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}