Dynamical phase transition in large-deviation statistics of the Kardar-Parisi-Zhang equation

We study the short-time behavior of the probability distribution P(H,t) of the surface height h(x = 0,t) = H in the Kardar-Parisi-Zhang (KPZ) equation in 1 + 1 dimension. The process starts from a stationary interface: h( x,t = 0) is given by a realization of two-sided Brownian motion constrained by h(0,0) = 0. We find a singularity of the large deviation function of H at a critical value H = H-c. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry x <-> -x of optimal paths h(x,t) predicted by the weak-noise theory of the KPZ equation. At vertical bar H vertical bar >> vertical bar H-c vertical bar the corresponding tail of P(II) scales as - ln P similar to vertical bar II vertical bar(3/2) / t(1/2) and agrees, at any t > 0, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of P scales as - ln P similar to vertical bar H vertical bar(5/2) / t(1/2) and coincides with the corresponding tail for the sharp-wedge initial condition.