Bootstrap Percolation and Partial Difference Equations

Abstract

We study a bootstrap percolation process on $\mathbb{Z}^d$ where each newly occupied point completes a copy of one of the patterns in some fixed collection $\mathcal{A}$. We aim, given $\mathcal{A}$, to find the smallest size of a percolating seed. We show that, for two patterns in two dimensions, in essence the answer is given by the mixed volume of the patterns' convex hulls. Our motivation for studying this setup comes from the theory of partial difference equations. Conversely, our considerations yield one consistency criterion for systems of two partial difference equations in two dimensions.