The Annals of Probability

Sublogarithmic fluctuations for internal DLA

Amine Asselah and Alexandre Gaudillière

Full-text: Open access

Abstract

We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random cluster growth model, where random walks start at the origin of the $d$-dimensional lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. It is known that the asymptotic shape of the cluster is a sphere. When the dimension is two or more, we have shown in a previous paper that the inner (resp., outer) fluctuations of its radius is at most of order $\log(\mathrm{radius})$ [resp., $\log^{2}(\mathrm{radius})$]. Using the same approach, we improve the upper bound on the inner fluctuation to $\sqrt{\log(\mathrm{radius})}$ when $d$ is larger than or equal to three. The inner fluctuation is then used to obtain a similar upper bound on the outer fluctuation.

Article information

Source
Ann. Probab., Volume 41, Number 3A (2013), 1160-1179.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1367241496

Digital Object Identifier
doi:10.1214/11-AOP735

Mathematical Reviews number (MathSciNet)
MR3098674

Zentralblatt MATH identifier
1274.60286

Subjects
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]

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

Citation

Asselah, Amine; Gaudillière, Alexandre. Sublogarithmic fluctuations for internal DLA. Ann. Probab. 41 (2013), no. 3A, 1160--1179. doi:10.1214/11-AOP735. https://projecteuclid.org/euclid.aop/1367241496


Export citation

References

  • [1] Asselah, A. and Gaudillière, A. (2013). From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models. Ann. Probab. 41 1115–1159.
  • [2] Jerison, D., Levine, L. and Sheffield, S. (2012). Logarithmic fluctuations for internal DLA. J. Amer. Math. Soc. 25 271–301.
  • [3] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
  • [4] Lawler, G. F. (1995). Subdiffusive fluctuations for internal diffusion limited aggregation. Ann. Probab. 23 71–86.
  • [5] Lawler, G. F., Bramson, M. and Griffeath, D. (1992). Internal diffusion limited aggregation. Ann. Probab. 20 2117–2140.
  • [6] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge.