Landing-limited aggregation and prime number detection

Abstract

An aggregation model is introduced based on the principles that N freely diffusing monomers stick together to form a cluster when they touch, the motion of clusters is directed, and limited by the requirement that a cluster be displaced over its entire size in one go if space permits. A successful displacement thus implies that the cluster lands on an unoccupied space adjacent to, and of the same size and shape as, the space which it occupied before the move. This “landing-limited aggregation” (LLA) can be motivated, for example, in the context of statistical mechanical modelling of the migration of animals herds, either self-organized in the wild or under human surveillance. This LLA is studied at fixed monomer density 1/2 on a periodic 1d lattice, or variations of it (staircase or double chain lattice), and on the 2d square lattice. In general the dynamics leads to three possible final states for the aggregate: a single cluster or “N-mer” containing the entire population, an arrested state in which multiple clusters are stuck in place, or multiple clusters that continue to move on the lattice without ever aggregating. On the 1d and the staircase lattices prime numbers play an interesting role in excluding certain final states. For example, for an initial condition on the periodic 1d lattice consisting of N monomers mutually separated by a vacant site, the final state never consists of just two clusters if and only if N is prime.