Weak-noise theory and large deviations of surface height in the Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang (KPZ) equation describes an important universality class of nonequilibrium stochastic models, including stochastic interface growth. There has been much recent interest in the one-point probability distribution P(H, t) of height H of the evolving interface at time t. I will show how one can use the weak-noise theory (also known as the optimal fluctuation method, the macroscopic fluctuation theory, or simply WKB) to evaluate P(H, t) for different initial conditions. At small t and H, P(H,t) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In a proper moving frame, one of the tails agrees at all times with the asymptotics of the Tracy-Widom distribution (for the flat and curved interface), and of the Baik-Rains distribution (for the stationary interface), previously observed at long times. The other tail displays a behavior that differs from the known long-time asymptotics. We argue that this tail should be also observable at long times once |H| is sufficiently large. The case of stationary interface is especially interesting. Here at short times the large deviation function of the height exhibits a singularity at a critical value of |H|. This singularity has the character of a second-order phase transition, and it results from a symmetry-breaking of the "optimal path" of the system.