Surface structure and anisotropy of Eden clusters

The simple Eden model is simulated with clusters which are orders of magnitude larger than those of some previous work. The 'surface' (perimeter) is slightly anisotropic and feels the underlying structure of the square lattice even for 17 million cluster sites. The width of the surface layer increases with increasing cluster mass, with an effective exponent presumably larger than 1/4 for averages over the whole circumference, and perhaps equal to 1/4 for Rat surfaces. Width averages over a small angle are compared with the recent prediction of Plischke and Racz. Even 17 million sites are not enough to see directly the asymptotic behaviour. The growth and forms of clusters [ 1,2], and among them those of Eden clusters [3], are of great current interest [4, 51. In the simplest version of the Eden cluster one starts from one occupied site as a seed on a lattice, and at every time step occupies one randomly selected perimeter site. Perimeters are the empty neighbours of the already occupied growth sites and can be called growth sites because of their kinetic implications. The growth sites in percolation [6] reduce to these growth sites only in the limit p-, 1. Since there are no such perimeter sites deep in the cluster interior [7,8], in the Eden model we do not have to distinguish between perimeter, surface, and growth sites, in contrast to more complicated models like crystal growth [9], percolation [lo]?, or Ising domain walls [ l l ]. We do not discuss here the possible biological implications and other applications of the Eden growth model but regard it as one of the most basic examples of clusters. For the same reason we restrict ourselves here to the aforementioned version of this model (model A of [5]) and do not consider the more complicated versions B and C of reference [5] even though they might be easier to investigate asymptotically. Most of our effort was put into Monte Carlo simulations on the square lattice, but we will also mention surface studies on a simple cubic lattice. The width W of the surface layer is defined through W* = (r')-(r)* and is expected to vary asymptotically with some power of the number s of sites in a cluster: W a s x (s-,m). Here r is the distance of a surface site from the origin (or centre of mass) of the cluster. Peters et a1 …