Abstract
In this paper we consider the Greenberg-Hastings and cyclic color models. These models exhibit (at least) three different types of behavior. Depending on the number of colors and the size of two parameters called the threshold and range, the Greenberg-Hastings model either dies out, or has equilibria that consist of “debris” or “fire fronts”. The phase diagram for the cyclic color models is more complicated. The main result of this paper, Theorem 1, proves that the debris phase exists for both systems.
Similar content being viewed by others
References
Bramson, M., and Griffeath, D. (1989). Flux and fixation in cyclic particle systems.Ann. Prob 17, 26–45.
Cox, J. T., and Durrett, R. (1991). Nonlinear voter models. InRandom Walks, Brownian Motion, and Interacting Particle Systems, eds. R. Dunett and H. Kesten, Birkhauser, Boston.
Durrett, R. (1984). Oriented percolation in two dimensions.Ann. Prob. 12, 999–1040.
Durrett, R., and Gray, L. (1985). Some peculiar properties of a particle system with sexual reproduction. Unpublished manuscript.
Durrett, R., and Griffeath, D. (1982). Contact processes in several dimensions.Z. Warsch. verw. Gebiete 59, 535–552.
Durrett, R., and Neuhauser, C. (1991). Epidemics with regrowth ind=2.Ann. Appl. Prob. 1, 189–206.
Durrett, R., and Steif, J. (1991). Some rigorous results for the Greenberg-Hastings model.J. Theor. Prob. 4, 669–690.
Fisch, R. (1990). Clustering in the one-dimensional 3-color cyclic cellular automaton.J. Theor. Prob. 3, 311–338.
Fisch, R., Gravner, J., and Griffeath, D. (1991). Cyclic cellular automata in two dimensions. InSpatial Stochastic Processes. A Festchrift in Honor of Ted Harris on his seventieth birthday, eds. K. Alexander, and J. Watkins, Birkhauser, Boston.
Fisch, R., Gravner, J., and Griffeath, D. (1991). Threshold-range scaling for excitable cellular automata.Statistics and Computing 1, 23–39.
Fisch. R., Gravner, J., and Griffeath, D. (1992). Metastability in the Greenberg-Hastings model. In preparation.
Greenberg, J. M., Hassard, B. D., and Hastings, S. P. (1978). Pattern formation and periodic structures in systems modeled by reaction-diffusion equations.Bull. AMS 84, 1296–1327.
Greenberg, J. M., and Hastings, S. P. (1978). Spatial patterns for discrete models of diffusion in excitable media.SIAM J. Appl. Math. 34, 515–523.
Kesten, H. (1986). Aspects of first passage percolation. InÉcole d'Été de Probabilités de Saint-Flour XIV. Springer Lecture Notes in Mathematics, Vol. 1180, Springer, New York.
Liggett, T. M. (1985).Interacting Particle Systems. Springer, New York.
Rosenblatt, M. (1965). Transition probability operators. In theFifth Berkeley Symposium, Vol. II, Part II, pp. 473–483.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Durrett, R. Multicolor particle systems with large threshold and range. J Theor Probab 5, 127–152 (1992). https://doi.org/10.1007/BF01046781
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01046781