Properties of Large Eden Clusters in the Plane

Abstract

Whereas the cylindrical version of an Eden cluster in the plane has a surface roughness with a fractal dimension predicted by theory, the central version has hitherto seemed to conflict with theory. However, a fresh way of analysing computer simulations of the central version shows that this anomaly is more apparent than real, and the central version can thereby be reconciled with theory. As a by-product, we obtain statistical data on the properties of the central version in the plane. The macroscopic shape of a central cluster is not circular, and microscopic roughness depends weakly upon the angular direction of portions of the surface. Rather surprisingly, the edge method of construction gives a more nearly circular shape than the external and internal methods. For higher dimensions than the plane, the corresponding treatment is more difficult, and there the situation remains obscure. Higher dimensions and certain other clusters (e.g.Richardson clusters) are treated briefly in Section 6. The theory of surface roughness uses a spatial generalization of martingales, called a serial harness: this is also described in Section 6.