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)=−|x|/δh(x,t=0)=−|x|/δ, δ≪1δ≪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)h(x,t) valid for any x   and t>0t>0. The corresponding distribution function converges in the long time limit, t→∞t→∞, to the Tracy–Widom distribution. The first order correction is a shift of order t−1/3t−1/3. We provide numerical computations based on the exact formula.