{
  "id": "2002.07760",
  "version": "v1",
  "published": "2020-02-18T17:49:28.000Z",
  "updated": "2020-02-18T17:49:28.000Z",
  "title": "Determinantal Point Processes, Stochastic Log-Gases, and Beyond",
  "authors": [
    "Makoto Katori"
  ],
  "comment": "LaTeX 77 pages, no figure. This manuscript has been prepared for the mini course given at `Workshop on Probability and Stochastic Processes' held at Orange County, Coorg, India, from 23rd to 26th February, 2020, which is organized by the Indian Academy of Sciences, Bangalore",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures, whose correlation functions are all given by determinants specified by an integral kernel called the correlation kernel. First we show our new scheme of DPPs in which a notion of partial isometies between a pair of Hilbert spaces plays an important role. Many examples of DPPs in one-, two-, and higher-dimensional spaces are demonstrated, where several types of weak convergence from finite DPPs to infinite DPPs are given. Dynamical extensions of DPP are realized in one-dimensional systems of diffusive particles conditioned never to collide with each other. They are regarded as one-dimensional stochastic log-gases, or the two-dimensional Coulomb gases confined in one-dimensional spaces. In the second section, we consider such interacting particle systems in one dimension. We introduce a notion of determinantal martingale and prove that, if the system has determinantal martingale representation (DMR), then it is a determinantal stochastic process (DSP) in the sense that all spatio-temporal correlation function are expressed by a determinant. In the last section, we construct processes of Gaussian free fields (GFFs) on simply connected proper subdomains of ${\\mathbb{C}}$ coupled with interacting particle systems defined on boundaries of the domains. There we use multiple Schramm--Loewner evolutions (SLEs) driven by the interacting particle systems. We prove that, if the driving processes are time-changes of the log-gases studied in the second section, then the obtained GFF with multiple SLEs are stationary. The stationarity defines an equivalence relation of GFFs, which will be regarded as a generalization of the imaginary surface studied by Miller and Sheffield.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-18T17:49:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "determinantal point process",
      "stochastic log-gases",
      "interacting particle systems",
      "determinantal martingale",
      "second section"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 77,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}