Diffusion-limited Aggregation Driven by Optimal Transportation

In this article, we combine the DLA model of Witten and Sander with ideas from optimal transportation. We propose a modification of the DLA model in which the probability of sticking is inversely proportional to the additional transport cost from the point to the root. We used a family of cost functions parametrized by a parameter α as studied in ramified optimal transportation. α < 0 promotes growth near the root whereas α > 0 promotes growth at the tips of the cluster. α = 0 is a phase transition point and corresponds to standard DLA. What makes this model interesting is that when α is negative enough (e.g. α < −2) the final cluster is an one dimensional curve. On the other hand, when α is positive enough (e.g. α > 2) we get a nearly two dimensional disk. Thus our model encompasses the full range of fractal dimension from 1 to 2.