Consider the short-time probability distribution $mathcal{P}(H,t)$ of the one-point interface height difference $h(x=0,tau=t)-h(x=0,tau=0)=H$ of the stationary interface $h(x,tau)$ described by the Kardar-Parisi-Zhang equation. It was previously shown that the optimal path -- the most probable history of the interface $h(x,tau)$ which dominates the higher tail of $mathcal{P}(H,t)$ -- is described by any of emph{two} ramp-like structures of $h(x,tau)$ traveling either to the left, or to the right. These two solutions emerge, at a critical value of $H$, via a spontaneous breaking of the mirror symmetry $xleftrightarrow-x$ of the optimal path, and this symmetry breaking is responsible for a second-order dynamical phase transition in the system. Here we employ a large-deviation Monte Carlo sampling algorithm in conjunction with the mapping between the KPZ interface and the directed polymer in a random potential at high temperature. This allows us to observe the optimal paths, which determine each of the two tails of $mathcal{P}(H,t)$, down to probability densities as small as $10^{-500}$. The observed mirror-symmetry-broken traveling optimal paths for the higher tail, and mirror-symmetric paths for the lower tail, are in good quantitative agreement with analytical predictions.
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