The Annals of Probability

From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

Amine Asselah and Alexandre Gaudillière

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We consider a cluster growth model on ${\mathbb{Z}}^{d}$, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension $d\ge2$. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachère in our Appendix) on the expected time spent by a random walk inside an annulus.

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Ann. Probab., Volume 41, Number 3A (2013), 1115-1159.

First available in Project Euclid: 29 April 2013

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B24: Interface problems; diffusion-limited aggregation 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Internal diffusion limited aggregation cluster growth random walk shape theorem logarithmic fluctuations subdiffusive fluctuations


Asselah, Amine; Gaudillière, Alexandre. From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models. Ann. Probab. 41 (2013), no. 3A, 1115--1159. doi:10.1214/12-AOP762.

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