Diffusion limited aggregation in the layers model

Abstract

In the classical model of Diffusion Limited Aggregation (DLA), introduced by Witten and Sander, the process begins with a single particle cluster placed at the origin of a space, and then, one at a time, particles make a random walk from infinity until they collide with, and stick to, the existing cluster. We consider an analogous version of this process on large but finite graphs with a designated source and sink vertex. Initially the cluster of halted particles contains a single particle at the sink vertex. Starting one at a time from the source, each particle makes a random walk in the direction of the sink vertex. The particle halts at the last unoccupied vertex before the walk enters the cluster for the first time, thus increasing the size of the cluster. This continues until the source vertex becomes occupied, at which point the process ends. We study the DLA process on several classes of layered graphs, including Cayley trees of branching factor at least two with a sink vertex attached to the leaves. We determine the finish time of the process for a given class of graphs and show that the subcomponent of the final cluster linking source to sink is essentially a unique path.