Logarithmic fluctuations for internal DLA

Jerison, David; Levine, Lionel; Sheffield, Scott

doi:10.1090/S0894-0347-2011-00716-9

Authors: David Jerison, Lionel Levine and Scott Sheffield
Journal: J. Amer. Math. Soc. 25 (2012), 271-301
MSC (2010): Primary 60G50, 60K35, 82C24
DOI: https://doi.org/10.1090/S0894-0347-2011-00716-9
Published electronically: August 15, 2011
MathSciNet review: 2833484
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let each of $n$ particles starting at the origin in $\mathbb Z^2$ perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set $A(n)$ of $n$ occupied sites is (with high probability) close to a disk $\mathbf {B}_r$ of radius $r=\sqrt {n/\pi }$. We show that the discrepancy between $A(n)$ and the disk is at most logarithmic in the radius: i.e., there is an absolute constant $C$ such that with probability $1$, \[ \mathbf {B}_{r - C\log r} \subset A(\pi r^2) \subset \mathbf {B}_{r+ C\log r} \quad \mbox { for all sufficiently large $r$}. \]

References

A. Asselah and A. Gaudillière, A note on the fluctuations for internal diffusion limited aggregation. arXiv:1004.4665

A. Asselah and A. Gaudillière, From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models. arXiv:1009.2838

M. Bramson and O. Zeitouni, Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. arXiv:1009.3443

P. Diaconis and W. Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Sem. Mat. Univ. Politec. Torino 49 (1991), no. 1, 95–119 (1993). Commutative algebra and algebraic geometry, II (Italian) (Turin, 1990). MR 1218674

T. Friedrich and L. Levine, Fast simulation of large-scale growth models. arXiv:1006.1003

Yasunari Fukai and Kôhei Uchiyama, Potential kernel for two-dimensional random walk, Ann. Probab. 24 (1996), no. 4, 1979–1992. MR 1415236, DOI https://doi.org/10.1214/aop/1041903213

Janko Gravner and Jeremy Quastel, Internal DLA and the Stefan problem, Ann. Probab. 28 (2000), no. 4, 1528–1562. MR 1813833, DOI https://doi.org/10.1214/aop/1019160497

D. Jerison, L. Levine and S. Sheffield, Internal DLA in higher dimensions. arXiv:1012.3453

D. Jerison, L. Levine and S. Sheffield, Internal DLA and the Gaussian free field. arXiv:1101.0596

Wouter Kager and Lionel Levine, Diamond aggregation, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 2, 351–372. MR 2670220, DOI https://doi.org/10.1017/S030500411000006X

Harry Kesten, Upper bounds for the growth rate of DLA, Phys. A 168 (1990), no. 1, 529–535. MR 1077203, DOI https://doi.org/10.1016/0378-4371%2890%2990405-H

Gady Kozma and Ehud Schreiber, An asymptotic expansion for the discrete harmonic potential, Electron. J. Probab. 9 (2004), no. 1, 1–17. MR 2041826, DOI https://doi.org/10.1214/EJP.v9-170

Gregory F. Lawler, Maury Bramson, and David Griffeath, Internal diffusion limited aggregation, Ann. Probab. 20 (1992), no. 4, 2117–2140. MR 1188055

Gregory F. Lawler, Subdiffusive fluctuations for internal diffusion limited aggregation, Ann. Probab. 23 (1995), no. 1, 71–86. MR 1330761

Gregory F. Lawler and Vlada Limic, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010. MR 2677157

Lionel Levine and Yuval Peres, Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Anal. 30 (2009), no. 1, 1–27. MR 2465710, DOI https://doi.org/10.1007/s11118-008-9104-6

Lionel Levine and Yuval Peres, Scaling limits for internal aggregation models with multiple sources, J. Anal. Math. 111 (2010), 151–219. MR 2747064, DOI https://doi.org/10.1007/s11854-010-0015-2

P. Meakin and J. M. Deutch, The formation of surfaces by diffusion-limited annihilation, J. Chem. Phys. 85:2320, 1986.

Cristopher Moore and Jonathan Machta, Internal diffusion-limited aggregation: parallel algorithms and complexity, J. Statist. Phys. 99 (2000), no. 3-4, 661–690. MR 1766912, DOI https://doi.org/10.1023/A%3A1018627008925

Peter Mörters and Yuval Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 30, Cambridge University Press, Cambridge, 2010. With an appendix by Oded Schramm and Wendelin Werner. MR 2604525

Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357

Scott Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), no. 3-4, 521–541. MR 2322706, DOI https://doi.org/10.1007/s00440-006-0050-1

David Bruce Wilson, Generating random spanning trees more quickly than the cover time, Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996) ACM, New York, 1996, pp. 296–303. MR 1427525, DOI https://doi.org/10.1145/237814.237880

T. A. Witten and L. M. Sander, Diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett. 47(19):1400–1403, 1981.