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

Abstract

We consider the KPZ equation in one space dimension with narrow wedge initial condition, h ( x , t = 0 ) = − | x | / δ , δ ≪ 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 → ∞ , 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.