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 language||American English|
|Journal||Journal of Statistical Mechanics: Theory and Experiment|
|State||Published - 8 Jun 2017|
Bibliographical noteFunding Information:
BM acknowledges financial support from the Israel Science Foundation (grant No. 807/16).
© 2017 IOP Publishing Ltd and SISSA Medialab srl.
- fluctuation phenomena
- interfaces in random media
- kinetic growth processes
- large deviations in non-equilibrium systems