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

Abstract

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

17 one-column pages, 8 figures, a few typos corrected

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Cited by in corpus (14)

• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions
• Short time large deviations of the KPZ equation
• Observing symmetry-broken optimal paths of stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media
• Lyapunov exponents of the SHE for general initial data
• Lyapunov exponents of the half-line SHE
• Integrability in the weak noise theory
• Upper tail decay of KPZ models with Brownian initial conditions
• KPZ fluctuations in finite volume
• Short-time large deviations of the spatially averaged height of a KPZ interface on a ring
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Large deviations for the KPZ equation from the KP equation
• Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth
• High moments of the SHE in the clustering regimes