Large deviations of surface height in the 1 + 1-dimensional Kardar-Parisi-Zhang equation: Exact long-time results for λh<0

Pavel Sasorov, Baruch Meerson, Sylvain Prolhac

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Abstract

We study atypically large fluctuations of height H in the 1 + 1-dimensional Kardar-Parisi-Zhang (KPZ) equation at long times t, when starting from a droplet initial condition. We derive exact large deviation function of height for λH < 0, where λ is the nonlinearity coefficient of the KPZ equation. This large deviation function describes a crossover from the Tracy-Widom distribution tail at small |H|/t, which scales as |H|3/t, to a different tail at large |H|/t, which scales as |H|5/2/t1/2. The latter tail exists at all times t > 0. It was previously obtained in the framework of the optimal fluctuation method. It was also obtained at short times from exact representation of the complete height statistics. The crossover between the two tails, at long times, occurs at |H| ∼ t as previously conjectured. Our analytical findings are supported by numerical evaluations using exact representation of the complete height statistics.

Original languageAmerican English
Article number063203
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2017
Issue number6
DOIs
StatePublished - 8 Jun 2017

Bibliographical note

Publisher Copyright:
© 2017 IOP Publishing Ltd and SISSA Medialab srl.

Keywords

  • fluctuation phenomena
  • interfaces in random media
  • kinetic growth processes
  • large deviations in non-equilibrium systems

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