The Annals of Probability

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

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 29 April 2013

https://projecteuclid.org/euclid.aop/1367241495

Digital Object Identifier
doi:10.1214/12-AOP762

Mathematical Reviews number (MathSciNet)
MR3098673

Zentralblatt MATH identifier
1283.60117

Citation

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. https://projecteuclid.org/euclid.aop/1367241495

References

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