Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation

Baruch Meerson, Eytan Katzav, Arkady Vilenkin
2016 Physical Review Letters  
Using the weak-noise theory, we evaluate the probability distribution $\mathcal{P}(H,t)$ of large deviations of height $H$ of the evolving surface height $h(x,t)$ in the Kardar-Parisi-Zhang (KPZ) equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height $H$ at time $t$. We argue that the tails of $\mathcal{P}$ behave, at arbitrary time $t>0$, and in a proper moving frame, as $-\ln \mathcal{P}\sim
more » ... \mathcal{P}\sim |H|^{5/2}$ and $\sim |H|^{3/2}$. The $3/2$ tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously observed at long times.
doi:10.1103/physrevlett.116.070601 pmid:26943523 fatcat:2cyttv5pzjcsdivu5lhbevxvq4