{
  "id": "2212.06777",
  "version": "v1",
  "published": "2022-12-13T17:52:01.000Z",
  "updated": "2022-12-13T17:52:01.000Z",
  "title": "Smooth statistics for a point process on the unit circle with reflection-type interactions across the real line",
  "authors": [
    "Christophe Charlier"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the point process \\begin{align*} \\frac{1}{Z_{n}}\\prod_{1 \\leq j < k \\leq n} |e^{i\\theta_{j}}-e^{-i\\theta_{k}}|^{\\beta}\\prod_{j=1}^{n} d\\theta_{j}, \\qquad \\theta_{1},\\ldots,\\theta_{n} \\in (-\\pi,\\pi], \\quad \\beta > 0, \\end{align*} where $Z_{n}$ is the normalization constant. The feature of this process is that the points $e^{i\\theta_{1}},\\ldots,e^{i\\theta_{n}}$ only interact with the mirror points reflected over the real line $e^{-i\\theta_{1}},\\ldots,e^{-i\\theta_{n}}$. We study smooth linear statistics of the form $\\sum_{j=1}^{n}g(\\theta_{j})$ as $n \\to \\infty$, where $g$ is $2\\pi$-periodic. We prove that a wide range of asymptotic scenarios can occur: depending on $g$, the leading order fluctuations around the mean can (i) be of order $n$ and purely Bernoulli, (ii) be of order $1$ and purely Gaussian, (iii) be of order $1$ and purely Bernoulli, or (iv) be of order $1$ and of the form $BN_{1}+(1-B)N_{2}$, where $N_{1},N_{2}$ are two independent Gaussians and $B$ is a Bernoulli that is independent of $N_{1}$ and $N_{2}$. The above list is not exhaustive: the fluctuations can be of order $n$, of order $1$ or $o(1)$, and other random variables can also emerge in the limit. We also obtain large $n$ asymptotics for $Z_{n}$ (and some generalizations), up to and including the term of order $1$. Our proof is inspired from a method developed by McKay and Wormald [15] to estimate related $n$-fold integrals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-13T17:52:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "point process",
      "real line",
      "unit circle",
      "smooth statistics",
      "reflection-type interactions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}