{
  "id": "1610.10080",
  "version": "v1",
  "published": "2016-10-31T19:34:40.000Z",
  "updated": "2016-10-31T19:34:40.000Z",
  "title": "Stochastic higher spin six vertex model and q-TASEPs",
  "authors": [
    "Daniel Orr",
    "Leonid Petrov"
  ],
  "comment": "AMSLaTeX; 45 pages, 13 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP",
    "math.QA"
  ],
  "abstract": "We present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the Macdonald processes hierarchy. First, we show how Macdonald $q$-difference operators with $t=0$ (an algebraic tool crucial for studying the corresponding Macdonald processes) can be utilized to get $q$-moments of the height function $\\mathfrak{h}$ in the higher spin six vertex model first computed in arXiv:1601.05770 using Bethe ansatz. This result in particular implies that for the vertex model with the step Bernoulli boundary condition, the value of $\\mathfrak{h}$ at an arbitrary point $(N+1,T)\\in\\mathbb{Z}_{\\ge2}\\times\\mathbb{Z}_{\\ge1}$ has the same distribution as the last component $\\lambda_N$ of a random partition under a specific $t=0$ Macdonald measure. On the other hand, it is known that $\\mathbf{x}_N:=\\lambda_N-N$ can be identified with the location of the $N$th particle in a certain discrete time $q$-TASEP started from the step initial configuration. The second construction we present is a coupling of this $q$-TASEP and the higher spin six vertex model (with the step Bernoulli boundary condition) along time-like paths providing an independent probabilistic explanation of the equality of $\\mathfrak{h}(N+1,T)$ and $\\mathbf{x}_N+N$ in distribution. Combined with the identification of averages of observables between the stochastic higher spin six vertex model and Schur measures (which are $t=q$ Macdonald measures) obtained recently in arXiv:1608.01553, this produces GUE Tracy--Widom asymptotics for a discrete time $q$-TASEP with the step initial configuration and special jump parameters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-31T19:34:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vertex model",
      "stochastic higher spin",
      "step bernoulli boundary condition",
      "step initial configuration",
      "produces gue tracy-widom asymptotics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}