  The competition interface between two growing ``Young clusters'' (diagrams),
in a two-dimensional random cone, is mapped to the path of a second-class
particle in the one-dimensional totally asymmetric simple exclusion process.
Using the asymptotics of the second class particle and hydrodynamic limits for
the exclusion process (Burgers equation), we show that the behavior of the
competition interface depends on the angle of the cone: for angles in [180^o,
270^o) the competition interface has a deterministic inclination, while for
angles in [90^o,180^o) the inclination is random. We relate the competition
model to a model of random directed polymers, and obtain some partial results
for the fluctuations of the competition interface.
