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For stationary interface growth, governed by the Kardar-Parisi-Zhang (KPZ)equation in 1 + 1 dimensions, typical fluctuations of the interface height atlong times are described by the Baik-Rains distribution. Recently Chhita et al.[1] used the totally asymmetric simple exclusion process (TASEP) to study theheight fluctuations in systems of the KPZ universality class for Brownianinterfaces with arbitrary diffusion constant. They showed that there is aone-parameter family of long-time distributions, parametrized by the diffusionconstant of the initial random height profile. They also computed thesedistributions numerically by using Monte Carlo (MC) simulations. Here weaddress this problem analytically and focus on the distribution tails at shorttimes. We determine the (stretched exponential) tails of the heightdistribution by applying the Optimal Fluctuation Method (OFM) to the KPZequation. We argue that, by analogy with other initial conditions, the "slow"tail holds at arbitrary times and therefore provides a proper asymptotic to thefamily of long-time distributions studied in Ref. [1]. We verify thishypothesis by performing large-scale MC simulations of a TASEP with aparallel-update rule. The "fast" tail, predicted by the OFM, is also expectedto hold at arbitrary times, at sufficiently large heights.

Height distribution tails in the Kardar-Parisi-Zhang equation with Brownian initial conditions