The boundary of iterates in Euclidean growth models
HTML articles powered by AMS MathViewer
- by Janko Gravner PDF
- Trans. Amer. Math. Soc. 348 (1996), 4549-4559 Request permission
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
- T. Bohman, work in preparation.
- Richard Durrett and David Griffeath, Asymptotic behavior of excitable cellular automata, Experiment. Math. 2 (1993), no. 3, 183–208. MR 1273408, DOI 10.1080/10586458.1993.10504277
- 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
- R. Fisch, J. Gravner, D. Griffeath, Threshold–range scaling for the excitable cellular automata, Statistic and Computing 1 (1991), 23–39.
- Janko Gravner and David Griffeath, Threshold growth dynamics, Trans. Amer. Math. Soc. 340 (1993), no. 2, 837–870. MR 1147400, DOI 10.1090/S0002-9947-1993-1147400-3
- J. Gravner, D. Griffeath, First passage times for discrete threshold growth dynamics, submitted to Ann. Prob. (1995).
- J. Gravner, D. Griffeath, Multitype threshold voter model and convergence to Poisson–Voronoi tessellation, preprint (1995).
- J. Gravner, Abstract growth dynamics, unpublished manuscript (1992).
- 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
- C. M. Newman, M. S. T. Piza, Divergence of shape fluctuations in two dimensions, Ann. Prob. 23 (1995), 977–1005.
- G. E. Pires, Threshold Growth Dynamics: a PDE Approach, Ph. D. Thesis, University of Wisconsin, Madison, 1995.
Additional Information
- Janko Gravner
- Affiliation: Department of Mathematics, University of California, Davis, California 95616
- Email: gravner@feller.ucdavis.edu
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 4549-4559
- MSC (1991): Primary 52A10; Secondary 52A99, 60K35
- DOI: https://doi.org/10.1090/S0002-9947-96-01697-2
- MathSciNet review: 1370643