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

We study the short-time behavior of the probability distribution \(\mathcal{P}(H,t)\) 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 \(x \leftrightarrow -x\) of optimal paths \(h(x,t)\) predicted by the weak-noise theory of the KPZ equation. At \(|H|\gg |H_c|\) the corresponding tail of \(\mathcal{P}(H)\) scales as \(-\ln \mathcal{P} \sim |H|^{3/2}/t^{1/2}\) 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 \(\mathcal{P}\) scales as \(-\ln \mathcal{P} \sim |H|^{5/2}/t^{1/2}\) and coincides with the corresponding tail for the sharp-wedge initial condition.