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We consider the KPZ equation in one space dimension with narrow wedge initialcondition, $h(x,t=0)=- |x|/\delta$, $\delta\ll 1$. Based on previous resultsfor the weakly asymmetric simple exclusion process with step initialconditions, we obtain a determinantal formula for the one-point distribution ofthe solution $h(x,t)$ valid for any $x$ and $t>0$. The correspondingdistribution function converges in the long time limit, $t\to\infty$, to theTracy-Widom distribution. The first order correction is a shift of order$t^{-1/3}$. We provide numerical computations based on the exact formula.

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