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