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Internal DLA in higher dimensions. (English) Zbl 1290.60051

The paper continues a research by the present authors, whose history is outlined in [J. Am. Math. Soc. 25, No. 1, 271–301 (2012; Zbl 1237.60037)]. Previously, a logarithmic estimate for internal DLA has been derived in \(d=2\). A number of modifications has been listed (like e.g. that of \(\log r\) by \(\sqrt{\log r}\)) that would necessarily appear in the \(d\geq 3\) extension of the theory. A key Lemma A of that paper has been formulated with an eye on that extension. The purpose of this note is to carry out in detail an adaptation of previous \(d=2\) techniques to higher dimensions. The ultimate outcome is a proof that for a cluster \(A(t)\) produced in the internal DLA with \(t\) particles in dimension \(d\geq 3\), its shape is spherical up to an \(O(\sqrt{\log t})\) error. Certain martingales related to the growth of the cluster are constructed and estimated. The estimates are closely parallel to what one obtains for the discrete Gaussian free field. This connection is elaborated in more detail in a forthcoming paper [the authors, “Internal DLA and the Gaussian free field”, arXiv:1101.0596].

MSC:

60G50 Sums of independent random variables; random walks
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics

Citations:

Zbl 1237.60037
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