One-dimensional long-range diffusion-limited aggregation I

Abstract

We examine diffusion-limited aggregation generated by a random walk on $\mathbb{Z}$ with long jumps. We derive upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. Under various regularity conditions on the tail of the step distribution, we prove that the diameter grows as $n^{\beta+o(1)}$, with an explicitly given $\beta$. The growth rate of the aggregate is shown to have three phase transitions, when the walk steps have finite third moment, finite variance, and conjecturally, finite half moment.