Internal DLA and the Gaussian free field

TL;DR

This paper investigates the average deviations of internal DLA clusters from their mean shape, revealing that fluctuations, when normalized, resemble a variant of the Gaussian free field, with results depending on the dimension.

Contribution

It introduces the concept of average error in internal DLA and shows that fluctuations scale to a Gaussian free field variant, extending previous shape results.

Findings

Average deviation is constant in 2D and scales as r^{1-d/2} in higher dimensions.

Normalized fluctuations converge to a Gaussian free field variant.

Shape deviations are smaller on average than maximal errors.

Abstract

In previous works, we showed that the internal DLA cluster on \Z^d with t particles is a.s. spherical up to a maximal error of O(\log t) if d=2 and O(\sqrt{\log t}) if d > 2. This paper addresses "average error": in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when d=2 and of order r^{1-d/2} (for a radius r cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.