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

TL;DR

This paper analyzes the probability of large height deviations in the 1D KPZ surface growth model using weak-noise theory, revealing tail behaviors that connect to Tracy-Widom distribution asymptotics.

Contribution

It provides a detailed evaluation of large deviation probabilities and optimal interface histories for the KPZ equation starting from flat initial conditions.

Findings

Tail behavior of probability distribution scales as |H|^{5/2} and |H|^{3/2}.

The 3/2 tail matches Tracy-Widom distribution asymptotics.

Results apply at arbitrary times in a moving frame.

Abstract

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 $-\ln \mathcal{P}\sim |H|^{5/2}$ 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.