## Abstract

We consider a stochastic interface h(x,t), described by the 1+1 Kardar-Parisi-Zhang (KPZ) equation on the half line x≥0 with the reflecting boundary at x=0. The interface is initially flat, h(x,t=0)=0. We focus on the short-time probability distribution P(H,L,t) of the height H of the interface at point x=L. Using the optimal fluctuation method, we determine the (Gaussian) body of the distribution and the strongly asymmetric non-Gaussian tails. We find that the slower-decaying tail scales as -tlnP≃|H|3/2f-(L/|H|t) and calculate the function f- analytically. Remarkably, this tail exhibits a first-order dynamical phase transition at a critical value of L, Lc=0.60223»|H|t. The transition results from a competition between two different fluctuation paths of the system. The faster-decaying tail scales as -tlnP≃|H|5/2f+(L/|H|t). We evaluate the function f+ using a specially developed numerical method which involves solving a nonlinear second-order elliptic equation in Lagrangian coordinates. The faster-decaying tail also involves a sharp transition which occurs at a critical value Lc≃22|H|t/π. This transition is similar to the one recently found for the KPZ equation on a ring, and we believe that it has the same fractional order, 5/2. It is smoothed, however, by small diffusion effects.

Original language | American English |
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Article number | 042132 |

Journal | Physical Review E |

Volume | 99 |

Issue number | 4 |

DOIs | |

State | Published - 22 Apr 2019 |

### Bibliographical note

Funding Information:We are grateful to Naftali Smith for useful discussions. T.A. and B.M. acknowledge financial support from the Israel Science Foundation (Grant No. 807/16).

Publisher Copyright:

© 2019 American Physical Society.