Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-08-19T05:41:43.421Z Has data issue: false hasContentIssue false
  • English
  • Français

Diamond aggregation

Published online by Cambridge University Press:  10 May 2010

WOUTER KAGER  and
LIONEL LEVINE
WOUTER KAGER
Affiliation:
Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. http://www.few.vu.nl/~wkager e-mail: wkager@few.vu.nl
LIONEL LEVINE
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.http://math.mit.edu/~levine e-mail: levine@math.mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Internal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 351 - 372
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[AS92]Alon, N. and Spencer, J. H.The probabilistic Method (John Wiley & Sons Inc., 1992).Google Scholar
[BQR03]Ben Arous, G., Quastel, J. and Ramírez, A. F.Internal DLA in a random environment. Ann. Inst. H. Poincaré Probab. Statist. 39 (2) (2003), 301–324.CrossRefGoogle Scholar
[Bi95]Billingsley, P.Probability and Measure, 3rd ed (John Wiley & Sons 1995).Google Scholar
[BB07]Blachère, S. and Brofferio, S.Internal diffusion limited aggregation on discrete groups having exponential growth. Probab. Theory Related Fields 137 (3–4) (2007), 323–343.CrossRefGoogle Scholar
[DF91]Diaconis, P. and Fulton, W.A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Politec. Torino 49 (1) (1991), 95–119.Google Scholar
[Du04]Dubédat, J.Reflected planar Brownian motions, intertwining relations and crossing probabilities. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), 539–552.CrossRefGoogle Scholar
[Es45]Esseen, C.-G.Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math. 77 (1945), 1–125.CrossRefGoogle Scholar
[GQ00]Gravner, J. and Quastel, J.Internal DLA and the Stefan problem. Ann. Probab. 28 (4) (2000), 1528–1562.CrossRefGoogle Scholar
[Ja02]Janson, S.On concentration of probability. In: Contemporary combinatorics. Bolyai Soc. Math. Stud. 10 (János Bolyai Math. Soc., Budapest 2002), 289–301.Google Scholar
[Ka07]Kager, W.Reflected Brownian motion in generic triangles and wedges. Stoch. Process. Appl. 117 (5) (2007), 539–549.CrossRefGoogle Scholar
[LBG92]Lawler, G. F., Bramson, M. and Griffeath, D.Internal diffusion limited aggregation. Ann. Probab. 20 (4) (1992), 2117–2140.CrossRefGoogle Scholar
[La95]Lawler, G. F.Subdiffusive fluctuations for internal diffusion limited aggregation. Ann. Probab. 23 (1) (1995), 71–86.CrossRefGoogle Scholar
[LP09a]Levine, L. and Peres, Y.Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile. Potential Anal. 30 (2009), 1–27.CrossRefGoogle Scholar
[LP09b]Levine, L. and Peres, Y. Scaling limits for internal aggregation models with multiple sources. J. d'Analyse Math., to appear. http://arxiv.org/abs/0712.3378.Google Scholar
[MM00]Moore, C. and Machta, J.Internal diffusion-limited aggregation: parallel algorithms and complexity. J. Statist. Phys. 99 (3–4) (2000), 661–690.CrossRefGoogle Scholar
[Pe66]Petrov, V. V.On a relation between an estimate of the remainder in the central limit theorem and the law of iterated logarithm. Theor. Probab. Appl. 11 (3) (1966), 454–458.CrossRefGoogle Scholar
[PS00]Prokhorov, Yu. V. and Statulevičius, V. (Eds.). Limit Theorems of Probability Theory (Springer-Verlag, 2000).CrossRefGoogle Scholar
[RP81]Rogers, L. C. G. and Pitman, J. W.Markov functions. Ann. Probab. 9 (4) (1981), 573–582.CrossRefGoogle Scholar