Exact short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation with Brownian initial condition.

Abstract

The early-time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension, starting from a Brownian initial condition with a drift w, is studied using the exact Fredholm determinant representation. For large drift we recover the exact results for the droplet initial condition, whereas a vanishingly small drift describes the stationary KPZ case, recently studied by weak noise theory (WNT). We show that for short time t, the probability distribution P(H,t) of the height H at a given point takes the large deviation form P(H,t)∼exp[-Φ(H)/sqrt[t]]. We obtain the exact expressions for the rate function Φ(H) for H<H_{c2}. Our exact expression for H_{c2} numerically coincides with the value at which WNT was found to exhibit a spontaneous reflection symmetry breaking. We propose two continuations for H>H_{c2}, which apparently correspond to the symmetric and asymmetric WNT solutions. The rate function Φ(H) is Gaussian in the center, while it has asymmetric tails, |H|^{5/2} on the negative H side and H^{3/2} on the positive H side.