The 'active perimeter' in cluster growth models: a rigorous bound

Abstract

A definition of an 'active perimeter' is introduced for arbitrary cluster growth models. A general bound is derived, showing that this perimeter cannot grow faster than R^A-1, where A=d_f is the fractal dimension of the cluster. For diffusion-limited aggregates, this is seen to be smaller than the 'growing interface' as defined by Meakin and Witten (1983). A further consequence is that the Eden model (1961) is fully compact in any dimension d, i.e. it has a negligible number of internal holes. Further one finds that its perimeter scales as (volume)^(d-1)d/.