{
  "id": "1312.2600",
  "version": "v2",
  "published": "2013-12-09T21:13:46.000Z",
  "updated": "2015-07-13T23:30:04.000Z",
  "title": "KPZ line ensemble",
  "authors": [
    "Ivan Corwin",
    "Alan Hammond"
  ],
  "comment": "96 pages, 11 figures",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For each t>1 we construct an N-indexed ensemble of random continuous curves with three properties: (1) The lowest indexed curve is distributed as the time t Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic PDE with narrow wedge initial data; (2) The entire ensemble satisfies a resampling invariance which we call the H-Brownian Gibbs property (with H(x)=e^{x}); (3) Increments of the lowest indexed curve, when centered by -t/24 and scaled down vertically by t^{1/3} and horizontally by t^{2/3}, remain uniformly absolutely continuous (i.e. have tight Radon-Nikodym derivatives) with respect to Brownian bridges as time t goes to infinity. This construction uses as inputs the diffusion that O'Connell discovered in relation to the O'Connell-Yor semi-discrete Brownian polymer, the convergence result of Moreno Flores-Quastel-Remenik of the lowest indexed curve of that diffusion to the solution of the KPZ equation with narrow wedge initial data, and the one-point distribution formula proved by Amir-Corwin-Quastel for the solution of the KPZ equation with narrow wedge initial data. We provide four main applications of this construction: (1) Uniform (as t goes to infinity) Brownian absolute continuity of the time t solution to the KPZ equation with narrow wedge initial data, even when scaled vertically by t^{1/3} and horizontally by t^{2/3}; (2) Universality of the t^{1/3} one-point (vertical) fluctuation scale for the solution of the KPZ equation with general initial data; (3) Concentration in the t^{2/3} scale for the endpoint of the continuum directed random polymer; (4) Exponential upper and lower tail bounds for the solution at fixed time of the KPZ equation with general initial data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-09T21:13:46.000Z",
      "comment": "94 pages, 11 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-07-13T23:30:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "narrow wedge initial data",
      "kpz equation",
      "kpz line ensemble",
      "lowest indexed curve",
      "general initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 96,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.2600C"
    }
  }
}