{
  "id": "2205.15785",
  "version": "v1",
  "published": "2022-05-31T13:39:00.000Z",
  "updated": "2022-05-31T13:39:00.000Z",
  "title": "Dynamical Loop Equation",
  "authors": [
    "Vadim Gorin",
    "Jiaoyang Huang"
  ],
  "comment": "92 pages, 10 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two--dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, $\\beta$--corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and $(q,\\kappa)$-distributions on lozenge tilings. Under technical assumptions, we show that the dynamical loop equations lead to Gaussian field type fluctuations. As an application, we compute the limit shape for $(q,\\kappa)$--distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian Free Field in an appropriate complex structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-31T13:39:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dynamical loop equation",
      "lozenge tilings",
      "gaussian field type fluctuations",
      "nonintersecting bernoulli/poisson random walks",
      "dyson brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 92,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}