We examine diffusion-limited aggregation generated by a random walk on 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 nβ+o(1), with an explicitly given β. 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.
Bibliographical notePublisher Copyright:
© Institute of Mathematical Statistics, 2016.
- Diffusion limited aggregation
- Green's function
- Harmonic measure
- Phase transition
- Random walk
- Stable Green's function
- Stable process