Limiting distributions for a polynuclear growth model with external sources

The purpose of this paper is to investigate the limiting distribution funct ions for a polynuclear growth model with two external sources which was con sidered by Prahofer and Spohn. Depending on the strength of the sources, th e limiting distribution functions are either the Tracy-Widom functions of r andom matrix theory or a new explicit function which has the special proper ty that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion proces s.