Models of cluster growth on the Cayley tree

Abstract

We study the diffusion-controlled process of cluster growth, introduced by Witten and Sander, on a Cayley tree. We show that it is then equivalent to the Eden model where growth occurs at any boundary site with equal probability. The mean level number and the square gyration radius of an $N$-particle aggregate both increase as $\left[\frac{K}{\mathrm{}(K-1\mathrm{})\mathrm{}}\right]\ln N$ on a tree of branching ratio $K$. The case of biased diffusion is studied numerically: an attractive bias does not change the logarithmic behavior of the size, but a repulsive bias leads to a different behavior, presumably with a mean level number of order $N$.