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

Understand

We consider the KPZ equation in one space dimension with narrow wedge initial condition, $h(x,t=0)=- |x|/\delta$, $\delta\ll 1$.

• 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, $t\to\infty$, to the Tracy-Widom distribution.
• The first order correction is a shift of order $t^{-1/3}$.