Maximal spanning time for neighborhood growth on the Hamming plane

TL;DR

This paper investigates the maximal time for a neighborhood growth process on a 2D lattice, providing bounds related to the shape of a defining Young diagram and analyzing specific initial conditions and simplified models.

Contribution

It introduces bounds on the maximal occupation time in a long-range growth model based on Young diagrams, advancing understanding of extremal growth dynamics on lattices.

Findings

Upper bound on growth time linear in diagram area

Lower bound proportional to square root of largest square side

Refined results for specific initial sets and simplified dynamics

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.