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.
Categories: math.CO