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Consider the short-time probability distribution $\mathcal{P}(H,t)$ of theone-point interface height difference $h(x=0,\tau=t)-h(x=0,\tau=0)=H$ of thestationary interface $h(x,\tau)$ described by the Kardar-Parisi-Zhang equation.It was previously shown that the optimal path -- the most probable history ofthe interface $h(x,\tau)$ which dominates the upper 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 acritical value of $H$, via a spontaneous breaking of the mirror symmetry$x\leftrightarrow -x$ of the optimal path, and this symmetry breaking isresponsible for a second-order dynamical phase transition in the system. Wesimulate the interface configurations numerically by employing alarge-deviation Monte Carlo sampling algorithm in conjunction with the mappingbetween the KPZ interface and the directed polymer in a random potential athigh temperature. This allows us to observe the optimal paths, which determineeach of the two tails of $\mathcal{P}(H,t)$, down to probability densities assmall as $10^{-500}$. At short times we observe mirror-symmetry-brokentraveling optimal paths for the upper tail, and a single mirror-symmetric pathfor the lower tail, in good quantitative agreement with analytical predictions.At long times, even at moderate values of $H$, where the optimal fluctuationmethod is \emph{not} supposed to apply, we still observe two well-defineddominating paths. Each of them violates the mirror symmetry $x\leftrightarrow-x$ and is a mirror image of the other.

physical review. e

Observing symmetry-broken optimal paths of stationary  Kardar-Parisi-Zhang interface via a large-deviation sampling of directed  polymers in random media