## Duke Mathematical Journal

### Internal DLA and the Gaussian free field

#### Abstract

In previous works, we showed that the internal diffusion-limited aggregation (DLA) cluster on $\mathbb {Z}^{d}$ with $t$ particles is almost surely spherical up to a maximal error of $O(\log t)$ if $d=2$ and $O(\sqrt{\log t})$ if $d\geq3$. 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.

#### Article information

Source
Duke Math. J., Volume 163, Number 2 (2014), 267-308.

Dates
First available in Project Euclid: 29 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1391007588

Digital Object Identifier
doi:10.1215/00127094-2430259

Mathematical Reviews number (MathSciNet)
MR3161315

Zentralblatt MATH identifier
1296.60113

#### Citation

Jerison, David; Levine, Lionel; Sheffield, Scott. Internal DLA and the Gaussian free field. Duke Math. J. 163 (2014), no. 2, 267--308. doi:10.1215/00127094-2430259. https://projecteuclid.org/euclid.dmj/1391007588

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