The Shape of the Limit Set in Richardson's Growth Model

Let Cₚ be the limiting shape of Richardson's growth model with parameter p (0, 1. Our main result is that if p is sufficiently close to one, then Cₚ has a flat edge. This means that Cₚ \x R²: x₁ + x₂ = 1\ is a nondegenerate interval. The value of p at which this first occurs is shown to be equal to the critical probability for a related contact process. For p < 1, we show that Cₚ is not the full diamond \x R²: \|x\| = |x₁| + |x₂| 1\. We also show that Cₚ is a continuous function of p, and that when properly rescaled, Cₚ converges as p 0 to the limiting shape for exponential site percolation.