Long-range one-dimensional internal diffusion-limited aggregation

TL;DR

This paper investigates the growth and shape of internal diffusion limited aggregation on the integer line, revealing how the cluster's structure depends on the increment distribution's moments and stable law domain of attraction.

Contribution

It extends previous results by establishing optimal moment conditions for cluster symmetry and shape in one-dimensional IDLA with general increment distributions.

Findings

Finite second moment leads to symmetric, contiguous clusters around the origin.

In the domain of attraction of a stable law with 1<α<2, clusters contain large contiguous blocks but may not be fully symmetric.

Results generalize prior work to broader classes of increment distributions.

Abstract

We study internal diffusion limited aggregation on $\mathbb{Z}$, where a cluster is grown incrementally by adding, for each random walk dispatched from the origin, the first site it reaches outside the cluster. We assume that the increment distribution $X$ of the driving random walks has $\mathbb{E} X =0$, but need neither be simple nor symmetric, and can have $\mathbb{E} (X^2) = \infty$, for example. For the case where $\mathbb{E} (X^2) < \infty$, we prove that after $m$ of the random walks have been dispatched, all but $o(m)$ sites in the cluster form an approximately symmetric contiguous block around the origin. This strengthens a result of Blach\`ere, for centred random walks whose increments have finite $3$rd moments, to the optimal moments condition. On the other hand, if $X$ is in the domain of attraction of a symmetric $\alpha$-stable law, $1 < \alpha <2$, we prove that the cluster contains a contiguous block of $\delta m +o(m)$ sites, where $0 < \delta < 1$, but, unlike the finite-variance case, one may not take $\delta=1$.