{
  "id": "2006.15934",
  "version": "v1",
  "published": "2020-06-29T11:06:02.000Z",
  "updated": "2020-06-29T11:06:02.000Z",
  "title": "Two-point convergence of the stochastic six-vertex model to the Airy process",
  "authors": [
    "Evgeni Dimitrov"
  ],
  "comment": "92 pages, 7 Figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time $T$, it is known that the one-point height function fluctuations are of order $T^{1/3}$ and governed by the Tracy-Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $T^{2/3}$ and vertically by $T^{1/3}$, converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin-Bufetov-Wheeler between the stochastic six-vertex model and the ascending Hall-Littlewood process (a certain measure on plane partitions). Using the Macdonald difference operators, we obtain formulas for two-point observables for the ascending Hall-Littlewood process, which for the six-vertex model give access to the joint cumulative distribution function for its height function. A careful asymptotic analysis of these observables gives the two-point convergence result under certain restrictions on the parameters of the model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-29T11:06:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B20",
      "60K35"
    ],
    "keywords": [
      "stochastic six-vertex model",
      "two-point convergence",
      "airy process",
      "ascending hall-littlewood process",
      "two-point distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 92,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}