  We study the competition interface between two growing clusters in a growth
model associated to last-passage percolation. When the initial unoccupied set
is approximately a cone, we show that this interface has an asymptotic
direction with probability 1. The behavior of this direction depends on the
angle $\theta$ of the cone: for $\theta\geq180^{\circ}$, the direction is
deterministic, while for $\theta<180^{\circ}$, it is random, and its
distribution can be given explicitly in certain cases. We also obtain partial
results on the fluctuations of the interface around its asymptotic direction.
The evolution of the competition interface in the growth model can be mapped
onto the path of a second-class particle in the totally asymmetric simple
exclusion process; from the existence of the limiting direction for the
interface, we obtain a new and rather natural proof of the strong law of large
numbers (with perhaps a random limit) for the position of the second-class
particle at large times.
