One-dimensional long-range diffusion-limited aggregation I

2009/10/23 by Gideon Amir, Omer Angel, Amir, Gideon +5
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.0910.4416

openalex publication_date 2009/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 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.

Citations

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