{
  "id": "1006.1338",
  "version": "v4",
  "published": "2010-06-07T19:16:40.000Z",
  "updated": "2013-05-24T12:58:37.000Z",
  "title": "Crossover distributions at the edge of the rarefaction fan",
  "authors": [
    "Ivan Corwin",
    "Jeremy Quastel"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/11-AOP725 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1003.0443",
  "journal": "Annals of Probability 2013, Vol. 41, No. 3A, 1243-1314",
  "doi": "10.1214/11-AOP725",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\\rho_-<\\rho_+$ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf--Cole solution of the Kardar-Parisi-Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466-537] and [Nuclear Phys. B 834 (2010) 523-542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian. We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as $T\\nearrow\\infty$ to those of the Airy $\\mathcal{A}_{2\\to \\mathrm{BM}}$ process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf-Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman-Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-05-24T12:58:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rarefaction fan",
      "crossover distributions",
      "step bernoulli initial data",
      "stochastic heat equation",
      "simple exclusion process"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.1338C"
    }
  }
}