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

Baruch Meerson, Johannes Schmidt

Research output: Contribution to journalArticlepeer-review


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 (2016 arXiv:1611.06690) 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, parameterized 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 longtime distributions studied in Chhita et al (2016 arXiv:1611.06690). We verify this hypothesis by performing large-scale MC simulations of a TASEP with a parallelupdate rule. The 'fast' tail, predicted by the OFM, is also expected to hold at arbitrary times, at sufficiently large heights.

Original languageAmerican English
Article number103207
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number10
StatePublished - 27 Oct 2017


  • exclusion processes
  • kinetic growth processes
  • large deviations in non-equilibrium systems
  • stochastic particle dynamics

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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