  Atypically large fluctuations in macroscopic non-equilibrium systems continue
to attract interest. Their probability can often be determined by the optimal
fluctuation method (OFM). The OFM brings about a conditional variational
problem, the solution of which describes the "optimal path" of the system which
dominates the contribution of different stochastic paths to the desired
statistics. The OFM proved efficient in evaluating the probabilities of rare
events in a host of systems. However, theoretically predicted optimal paths
were observed in stochastic simulations only in diffusive lattice gases, where
the predicted optimal density patterns are either stationary, or travel with
constant speed. Here we focus on the one-point height distribution of the
paradigmatic Kardar-Parisi-Zhang interface. Here the optimal paths,
corresponding to the distribution tails at short times, are intrinsically
non-stationary and can be predicted analytically. Using the mapping to the
directed polymer in a random potential at high temperature, we obtain
"snapshots" of the optimal paths in Monte-Carlo simulations which probe the
tails with an importance sampling algorithm. For each tail we observe a very
narrow "tube" of height profiles around a single optimal path which agrees with
the analytical prediction. The agreement holds even at long times, supporting
earlier assertions of the validity of the OFM in the tails well beyond the
weak-noise limit.
