Exact height distributions for the KPZ equation with narrow wedge initial condition

We consider the KPZ equation in one space dimension with narrow wedge initial condition, h(x, t = 0) = -vertical bar x vertical bar/delta, delta << 1, evolving into a parabolic profile with superimposed fluctuations. Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we obtain a determinantal formula for the one-point distribution of the solution h(x, t) valid for any x and t > 0. The corresponding distribution function converges in the long time limit, t -> infinity, to the Tracy-Widom distribution. The first order correction is a shift of order t(-1/3). We provide numerical computations based on the exact formula. (C) 2010 Elsevier B.V. All rights reserved.