Improved Upper Bounds on the Asymptotic Growth Velocity of Eden Clusters

TL;DR

This paper presents two refined methods to improve the upper bounds on the asymptotic growth velocity of Eden clusters on hypercubic lattices, with results varying across different dimensions.

Contribution

It introduces two novel modifications to the existing bounds on Eden cluster growth velocity, enhancing accuracy especially in high dimensions.

Findings

Bounds are asymptotically exact as dimension increases.

In two dimensions, the improvement is only a few percent.

The methods provide tighter upper bounds on growth velocity.

Abstract

We consider the asymptotic shape of clusters in the Eden model on a d-dimensional hypercubical lattice. We discuss two improvements for the well-known upper bound to the growth velocity in different directions by that of the independent branching process (IBP). In the IBP, each cell gives rise to a daughter cell at a neighboring site at a constant rate. In the first improvement, we do not allow such births along the bond connecting the cell to its mother cell. In the second, we iteratively evolve the system by a growth as IBP for a duration $\Delta$ t, followed by culling process in which if any cell produced a descendant within this interval, who occupies the same site as the cell itself, then the descendant is removed. We study the improvement on the upper bound on the velocity for different dimensions d. The bounds are asymptotically exact in the large-d limit. But in $d =2$, the improvement over the IBP approximation is only a few percent.