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Transactions of the American Mathematical Society

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The boundary of iterates in
Euclidean growth models

Author: Janko Gravner
Journal: Trans. Amer. Math. Soc. 348 (1996), 4549-4559
MSC (1991): Primary 52A10; Secondary 52A99, 60K35
MathSciNet review: 1370643
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Abstract: This paper defines a general Euclidean growth model via a translation invariant, monotone and local transformation on Borel subsets of $\mathbf {R}^2$. The main result gives a geometric condition for the boundary curvature of the iterates to go to 0. Consequences include estimates for the speed of convergence to the asymptotic shape, and a result about survival of Euclidean deterministic forest fires.

References [Enhancements On Off] (What's this?)

  • [Boh] T. Bohman, work in preparation.
  • [DG] Richard Durrett and David Griffeath, Asymptotic behavior of excitable cellular automata, Experiment. Math. 2 (1993), no. 3, 183–208. MR 1273408
  • [Dur] Richard Durrett, Lecture notes on particle systems and percolation, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1988. MR 940469
  • [FGG] R. Fisch, J. Gravner, D. Griffeath, Threshold--range scaling for the excitable cellular automata, Statistic and Computing 1 (1991), 23--39.
  • [GG1] Janko Gravner and David Griffeath, Threshold growth dynamics, Trans. Amer. Math. Soc. 340 (1993), no. 2, 837–870. MR 1147400,
  • [GG2] J. Gravner, D. Griffeath, First passage times for discrete threshold growth dynamics, submitted to Ann. Prob. (1995).
  • [GG3] J. Gravner, D. Griffeath, Multitype threshold voter model and convergence to Poisson--Voronoi tessellation, preprint (1995).
  • [Gra] J. Gravner, Abstract growth dynamics, unpublished manuscript (1992).
  • [Gri] David Griffeath, Self-organization of random cellular automata: four snapshots, Probability and phase transition (Cambridge, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 420, Kluwer Acad. Publ., Dordrecht, 1994, pp. 49–67. MR 1283175
  • [NP] C. M. Newman, M. S. T. Piza, Divergence of shape fluctuations in two dimensions, Ann. Prob. 23 (1995), 977--1005. CMP 95:17
  • [Pir] G. E. Pires, Threshold Growth Dynamics: a PDE Approach, Ph. D. Thesis, University of Wisconsin, Madison, 1995.

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Additional Information

Janko Gravner
Affiliation: Department of Mathematics, University of California, Davis, California 95616

Keywords: Growth dynamics, curvature, deterministic forest fire
Received by editor(s): July 14, 1995
Additional Notes: This research was partially supported by the research grant J1-6157-0101-94 from the Republic of Slovenia’s Ministry of Science
Article copyright: © Copyright 1996 American Mathematical Society