• DOI:10.1103/PhysRevE.94.032108
• Corpus ID: 118604131

Short-time height distribution in 1d KPZ equation: starting from a parabola

@inproceedings{Kamenev2016ShorttimeHD,
  title={Short-time height distribution in 1d KPZ equation: starting from a parabola},
  author={Alex Kamenev and Baruch Meerson and Pavel Sasorov},
  year={2016}
}

• A. Kamenev, B. Meerson, P. Sasorov
• Published 19 May 2016
• Mathematics

We study the probability distribution P(H, t, L) of the surface height h(x = 0, t) = H in the Kardar-Parisi-Zhang (KPZ) equation in 1 + 1 dimension when starting from a parabolic interface, h(x, t = 0) = x/L. The limits of L → ∞ and L → 0 have been recently solved exactly for any t > 0. Here we address the early-time behavior of P(H, t, L) for general L. We employ the weak-noise theory a variant of WKB approximation – which yields the optimal history of the interface, conditioned on reaching the given height H at the origin at time t. We find that at small H P(H, t, L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as − lnP = f+|H | /t and f−|H | /t. The factor f+(L, t) monotonically increases as a function of L, interpolating between time-independent values at L = 0 and L = ∞ that were previously known. The factor f− is independent of L and t, signalling universality of this tail for a whole class of deterministic initial conditions.