Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case

Abstract

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.