One-dimensional long-range diffusion-limited aggregation I

TL;DR

This paper investigates the growth dynamics of one-dimensional long-range diffusion-limited aggregation driven by a random walk with heavy-tailed jumps, establishing bounds and phase transitions based on moment conditions.

Contribution

It provides explicit growth rate bounds and identifies phase transitions in aggregate growth depending on the tail behavior of the step distribution.

Findings

Growth rate of the aggregate is n^{beta+o(1)} with explicit beta.

Three phase transitions occur at finite third moment, finite variance, and conjecturally finite half moment.

Derived bounds depend on the tail regularity of the walk's step distribution.

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.