The one-dimensional Kardar-Parisi -Zhang equation and its universality class

In 1986 Kardar, Parisi, and Zhang proposed a stochastic PDE for the motion of driven interfaces,
in particular for growth processes with local updating rules. The solution to the 1D KPZ equation
can be approximated through the weakly asymmetric simple exclusion process. Based on work of 
Tracy and Widom on the PASEP, we obtain an exact formula for the one-point generating function of the KPZ
equation in case of sharp wedge initial data. Our result is valid for all times, but of particular interest is
the long time behavior, related to random matrices, and the finite time corrections. This is joint work with 
Tomohiro Sasamoto.