Two-dimensional supercritical growth dynamics with one-dimensional nucleation

Abstract

We introduce a class of cellular automata growth models on the two-dimensional integer lattice with finite cross neighborhoods. These dynamics are determined by a Young diagram $\mathcal{Z}$ and the radius $\rho$ of the neighborhood, which we assume to be sufficiently large. A point becomes occupied if the pair of counts of currently occupied points on the horizontal and vertical parts of the neighborhood lies outside $\mathcal{Z}$. Starting with a small density $p$ of occupied points, we focus on the first time $T$ at which the origin is occupied. We show that $T$ scales as a power of $1/p$, and identify that power, when $\mathcal{Z}$ is the triangular set that gives threshold-$r$ bootstrap percolation, when $\mathcal{Z}$ is a rectangle, and when it is a union of a finite rectangle and an infinite strip. We give partial results when $\mathcal{Z}$ is a union of two finite rectangles. The distinguishing feature of these dynamics is nucleation of lines that grow to significant length before most of the space is covered.

35 pages