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

Abstract

Let $C_p$ be the limiting shape of Richardson's growth model with parameter $p \in (0, 1\rbrack$. Our main result is that if $p$ is sufficiently close to one, then $C_p$ has a flat edge. This means that $\partial C_p \cap \{x \in R^2:x_1 + x_2 = 1\}$ 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 $C_p$ is not the full diamond $\{x \in R^2:\|x\| = |x_1| + |x_2| \leq 1\}$. We also show that $C_p$ is a continuous function of $p$, and that when properly rescaled, $C_p$ converges as $p \rightarrow 0$ to the limiting shape for exponential site percolation.