Roughening and inclination of competition interfaces

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.