Abstract. We consider some variants of DLA which shift the distribution of the place where a new particle is added in a very strong way to the points of maximal harmonic measure. As a consequence these variants can grow like “generalized plus signs”, with the aggregate containing only points on the coordinate axes at all times.

Introduction and statement of results.

We construct connected lattice sets An, n = 1, 2, …, by two procedures, both of which are variants of common procedures in DLA (Diffusion Limited Aggregation). The original DLA model was introduced by Witten and Sander [22]; see also [13] and [21, Sect. 6] for a general introduction to DLA. We only consider lattice models, so that An is a connected subset of ℤd. We always take A1 = {0} and An will contain exactly n sites. The site added to An to make An+1 is denoted by yn, so that An+1 = An ∪ {yn}. yn is chosen from ∂An, the boundary of An, which is the collection of sites adjacent to <An, but not in An. To describe the distribution of yn we introduce some notation. Let Sk, K ≥ 0, be a simple symmetric nearest neighbor random walk on ℤd.