{
  "id": "1903.04945",
  "version": "v1",
  "published": "2019-03-12T14:22:29.000Z",
  "updated": "2019-03-12T14:22:29.000Z",
  "title": "Partial Isometries, Duality, and Determinantal Point Processes",
  "authors": [
    "Makoto Katori",
    "Tomoyuki Shirai"
  ],
  "comment": "AMS-LaTeX, 49 pages, no figure",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures $\\Xi$ on a space $S$ with measure $\\lambda$, whose correlation functions are all given by determinants specified by an integral kernel $K$ called the correlation kernel. We consider a pair of Hilbert spaces, $H_{\\ell}, \\ell=1,2$, which are assumed to be realized as $L^2$-spaces, $L^2(S_{\\ell}, \\lambda_{\\ell})$, $\\ell=1,2$, and introduce a bounded linear operator ${\\cal W} : H_1 \\to H_2$ and its adjoint ${\\cal W}^{\\ast} : H_2 \\to H_1$. We prove that if both of ${\\cal W}$ and ${\\cal W}^{\\ast}$ are partial isometries and both of ${\\cal W}^{\\ast} {\\cal W}$ and ${\\cal W} {\\cal W}^{\\ast}$ are of locally trace class, then we have unique pair of DPPs, $(\\Xi_{\\ell}, K_{\\ell}, \\lambda_{\\ell})$, $\\ell=1,2$, which satisfy useful duality relations. We assume that ${\\cal W}$ admits an integral kernel $W$ on $L^2(S_1, \\lambda_1)$, and give practical setting of $W$ which makes ${\\cal W}$ and ${\\cal W}^{\\ast}$ satisfy the above conditions. In order to demonstrate that the class of DPPs obtained by our method is large enough to study universal structures in a variety of DPPs, we show many examples of DPPs in one-, two-, and higher-dimensional spaces $S$, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ($d \\in \\mathbb{N}$) series of infinite DPPs on $S=\\mathbb{R}^d$ and $\\mathbb{C}^d$ are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-12T14:22:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "60B20",
      "46E22",
      "60B10"
    ],
    "keywords": [
      "determinantal point process",
      "partial isometries",
      "integral kernel",
      "infinite dpps",
      "satisfy useful duality relations"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}