Time-averaged height distribution of the Kardar-Parisi-Zhang interface

Naftali R. Smith, Baruch Meerson, Arkady Vilenkin

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We study the complete probability distribution P H , t of the time-averaged height H = (1/t) t 0 h(x = 0, t) dt at point x = 0 of an evolving 1 + 1 dimensional Kardar Parisi Zhang (KPZ) interface h (x, t). We focus on short times and flat initial condition and employ the optimal fluctuation method to determine the variance and the third cumulant of the distribution, as well as the asymmetric stretched-exponential tails. The tails scale as ?lnP H 3/2 /t and ?lnP H 5/2 /t, similarly to the previously determined tails of the one-point KPZ height statistics at specified time t = t. The optimal interface histories, dominating these tails, are markedly di erent. Remarkably, the optimal history, h (x = 0, t), of the interface height at x = 0 is a non-monotonic function of time: the maximum (or minimum) interface height is achieved at an intermediate time. We also address a more general problem of determining the probability density of observing a given height history of the KPZ interface at point x = 0.

Original languageAmerican English
Article number53207
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number5
StatePublished - 30 May 2019

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© 2019 Institute of Physics Publishing. All rights reserved.


  • Fluctuation phenomena
  • Kinetic growth processes
  • Large deviations in non-equilibrium systems


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