Critical one-dimensional multi-particle DLA

Abstract:   In multi-particle Diffusion Limited Aggregation (DLA) a sea of particles perform independent random walks until they run into the aggregate and are absorbed. In dimension 1, the rate of growth of the aggregate depends on \lambda, the density of the particles. Kesten and Sidoravicius proved that when \lambda<1 the aggregate grows like t^{1/2}. They furthermore predicted linear growth when \lambda>1 (subsequently confirmed) and t^{2/3} growth at the critical density \lambda=1. We address the critical case, confirming the t^{2/3} rate of growth and show that aggregate has a scaling limit whose derivative is a (–1/3)-self-similar diffusion process. Surprisingly, this leads to a slower-than-expected speed in the mildly supercritical regime when \lambda=1+\epsilon.