Large fluctuations of a Kardar-Parisi-Zhang interface on a half line: The height statistics at a shifted point

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.