Maximal spanning time for neighborhood growth on the Hamming plane

Abstract

We consider a long-range growth dynamics on the two-dimensional integer lattice, initialized by a finite set of occupied points. Subsequently, a site $x$ becomes occupied if the pair consisting of the counts of occupied sites along the entire horizontal and vertical lines through $x$ lies outside a fixed Young diagram $\mathcal{Z}$. We study the extremal quantity $\mu(\mathcal{Z})$, the maximal finite time at which the lattice is fully occupied. We give an upper bound on $\mu(\mathcal{Z})$ that is linear in the area of the bounding rectangle of $\mathcal{Z}$, and a lower bound $\sqrt{s-1}$, where $s$ is the side length of the largest square contained in $\mathcal{Z}$. We give more precise results for a restricted family of initial sets, and for a simplified version of the dynamics.