Isotropic model for cluster growth on a regular lattice

Abstract

There exists a plethora of mathematical models for cluster growth and/or aggregation on regular lattices. Almost all suffer from inherent anisotropy caused by the regular lattice upon which they are grown. We analyze the little-known model for stochastic cluster growth on a regular lattice first introduced by Ferreira Jr. and Alves [J. Stat. Mech. Theo. & Exp. (2006) P11007], which produces circular clusters with no discernible anisotropy. We demonstrate that even in the noise-reduced limit the clusters remain circular. We adapt the model by introducing a specific rearrangement algorithm so that, rather than adding elements to the cluster from the outside (corresponding to apical growth), our model uses mitosis-like cell splitting events to increase the cluster size. We analyze the surface scaling properties of our model and compare it to the behavior of more traditional models. In "1+1" dimensions we discover and explore a new, nonmonotonic surface thickness scaling relationship which differs significantly from the Family-Vicsek scaling relationship. This suggests that, for models whose clusters do not grow through particle additions which are solely dependent on surface considerations, the traditional classification into "universality classes" may not be appropriate.