A Convergence Rate for Extended-Source Internal DLA in the Plane

TL;DR

This paper analyzes the convergence rate of an extended-source internal DLA model in the plane, showing that fluctuations are bounded by a power of the lattice size with high probability, advancing understanding of its scaling behavior.

Contribution

It provides the first quantitative convergence rate for extended-source IDLA in the plane, linking fluctuations to lattice size with probabilistic bounds.

Findings

Fluctuations are at most of order δ^{3/5} from the scaling limit.

Convergence probability exceeds 1 - e^{-1/δ^{2/5}}.

Results quantify the speed of convergence for extended-source IDLA.

Abstract

Internal DLA (IDLA) is an internal aggregation model in which particles perform random walks from the origin, in turn, and stop upon reaching an unoccupied site. Levine and Peres showed that, when particles start instead from fixed multiple-point distributions, the modified IDLA processes have deterministic scaling limits related to a certain obstacle problem. In this paper, we investigate the convergence rate of this "extended source" IDLA in the plane to its scaling limit. We show that, if $\delta$ is the lattice size, fluctuations of the IDLA occupied set are at most of order $\delta^{3/5}$ from its scaling limit, with probability at least $1-e^{-1/\delta^{2/5}}$.