{
  "id": "1904.13371",
  "version": "v1",
  "published": "2019-04-30T17:08:18.000Z",
  "updated": "2019-04-30T17:08:18.000Z",
  "title": "A hierarchy of Palm measures for determinantal point processes with gamma kernels",
  "authors": [
    "Alexander I. Bufetov",
    "Grigori Olshanski"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.FA",
    "math.MP"
  ],
  "abstract": "The gamma kernels are a family of projection kernels $K^{(z,z')}=K^{(z,z')}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Euler's gamma function and depend on two continuous parameters $z,z'$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $\\mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z')}$ serves as a correlation kernel for a determinantal measure $M^{(z,z')}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$ \\ldots, K^{(z-1,z'-1)}, \\; K^{(z,z')},\\; K^{(z+1,z'+1)}, \\ldots, $$ and establish the following hierarchical relations inside any such chain: Given $(z,z')$, the kernel $K^{(z,z')}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z'+1)}$, and the one-point Palm distributions for the measure $M^{(z,z')}$ are absolutely continuous with respect to $M^{(z+1,z'+1)}$. We also explicitly compute the corresponding Radon-Nikod\\'ym derivatives and show that they are given by certain normalized multiplicative functionals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-30T17:08:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "determinantal point processes",
      "palm measures",
      "gamma kernels initially arose",
      "infinite point configurations",
      "eulers gamma function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}