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

We consider the KPZ equation in one space dimension with narrow wedge initial condition, image, image, 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 image valid for any x and image. The corresponding distribution function converges in the long time limit, image, to the Tracy–Widom distribution. The first order correction is a shift of order image. We provide numerical computations based on the exact formula.