Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition

We consider the relaxation (noise-free) statistics of the one-point height $H=h(x=0,t)$ where $h(x,t)$ is the evolving height of a one-dimensional Kardar-Parisi-Zhang (KPZ) interface, starting from a Brownian (random) initial condition. We find that, at short times, the distribution of $H$ takes the same scaling form $-\ln\mathcal{P}\left(H,t\right)=S\left(H\right)/\sqrt{t}$ as the distribution of H for the KPZ interface driven by noise, and we find the exact large-deviation function $S(H)$ analytically. At a critical value $H=H_c$, the second derivative of $S(H)$ jumps, signaling a dynamical phase transition (DPT). Furthermore, we calculate exactly the most likely history of the interface that leads to a given $H$, and show that the DPT is associated with spontaneous breaking of the mirror symmetry $x \leftrightarrow -x$ of the interface. In turn, we find that this symmetry breaking is a consequence of the non-convexity of a large-deviation function that is closely related to $S(H)$, and describes a similar problem but in half space. Moreover, the critical point $H_c$ is related to the inflection point of the large-deviation function of the half-space problem.