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 -root t ln P similar or equal to vertical bar H vertical bar(3/2)f-(L/root vertical bar H vertical bar t) and calculate the function f analytically. Remarkably, this tail exhibits a first-order dynamical phase transition at a critical value of L, L-c = 0.602 23 . . .root vertical bar H vertical bar t. The transition results from a competition between two different fluctuation paths of the system. The faster-decaying tail scales as -root t ln P similar or equal to vertical bar H vertical bar(5/2)f(+) (L/root vertical bar H vertical bar 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 L-c similar or equal to 2 root 2 vertical bar H vertical bar t/pi. 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.