{
  "id": "1605.06130",
  "version": "v1",
  "published": "2016-05-19T20:15:42.000Z",
  "updated": "2016-05-19T20:15:42.000Z",
  "title": "Short-time height distribution in 1d KPZ equation: starting from a parabola",
  "authors": [
    "Alex Kamenev",
    "Baruch Meerson",
    "Pavel V. Sasorov"
  ],
  "comment": "9 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We study the probability distribution $\\mathcal{P}(H,t,L)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension when starting from a parabolic interface, $h(x,t=0)=x^2/L$. The limits of $L\\to\\infty$ and $L\\to 0$ have been recently solved exactly for any $t>0$. Here we address the early-time behavior of $\\mathcal{P}(H,t,L)$ for general $L$. We employ the weak-noise theory - a variant of WKB approximation -- which yields the optimal history of the interface, conditioned on reaching the given height $H$ at the origin at time $t$. We find that, at small $H$, $\\mathcal{P}(H,t,L)$ is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as $-\\ln \\mathcal{P}= f_{+}|H|^{5/2}/t^{1/2}$ and $f_{-}|H|^{3/2}/t^{1/2}$. The factor $f_{+}(L,t)$ monotonically increases as a function of $L$, interpolating between time-independent values at $L=0$ and $L=\\infty$ that were previously known. The factor $f_{-}$ is independent of $L$ and $t$, signalling universality of this tail for a whole class of deterministic initial conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-19T20:15:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "1d kpz equation",
      "short-time height distribution",
      "deterministic initial conditions",
      "early-time behavior",
      "weak-noise theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}