{
  "id": "1803.05887",
  "version": "v1",
  "published": "2018-03-15T17:40:41.000Z",
  "updated": "2018-03-15T17:40:41.000Z",
  "title": "Coulomb-gas electrostatics controls large fluctuations of the KPZ equation",
  "authors": [
    "Ivan Corwin",
    "Promit Ghosal",
    "Alexandre Krajenbrink",
    "Pierre Le Doussal",
    "Li-Cheng Tsai"
  ],
  "comment": "6 pages + 7 pages of supplementary material; 2 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process -- an infinite particle Coulomb-gas which arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straight-forward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution which demonstrate a crossover between exponential decay with exponent $3$ (in the shallow left tail) to exponent $5/2$ (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov et al. [J. Stat. Mech, 063203 (2017)] via a WKB approximation analysis of a non-local, non-linear integro-differential equation generalizing Painlev\\'e II which Amir et al. [Comm. Pure Appl. Math. 64, 466 (2011)] related to the KPZ one-point distribution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-15T17:40:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "coulomb-gas electrostatics controls large fluctuations",
      "kpz equation",
      "large deviation rate function",
      "integro-differential equation generalizing painleve",
      "left tail"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}