Summary
This paper is a sequel of a paper of Cox and Griffeath “diffusive clustering in the two dimensional voter model”. We continue our study of the voter model and coalescing random walks on the two dimensional integer lattice. Some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived. We use these results to prove a limit law, announced in that earlier paper, concerning the size of the largest square centered at the origin which is of solid color at a large time t.
References
Bramson, M., Griffeath, D.: Asymptotics for interacting particle systems on \(\mathbb{Z}^{\text{d}} \). Z. Wahrscheinlichkeitstheor. Verw. Geb. 45, 183–196 (1980)
Cox, J.T., Griffeath, D.: Diffusive clustering in the two dimensional voter model. Ann. Probab. 14, 347–370 (1986)
Griffeath, D.: Additive and cancellative interacting particle systems. Lecture Notes Math. 724. Berlin-Heidelberg-New York: Springer 1979
Holley, R., Stroock, D.: Central limit phenomena of various interacting systems. Ann. Math. 110, 333–393 (1979)
Liggett, T.M.: Interacting particle systems. Berlin-Heidelberg-New York: Springer 1985
Tavaré, S.: Line-of-descent and geneological processes, and their applications in population genetics models. Theor. Popul. Biol. 26, 119–164 (1984)
Author information
Authors and Affiliations
Additional information
Partially supported by the National Science Foundation under Grant DMS-831080
Partially supported by the National Science Foundation under Grant DMS-841317
Partially supported by the National Science Foundation under Grant DMS-830549
Rights and permissions
About this article
Cite this article
Bramson, M., Cox, J.T. & Griffeath, D. Consolidation rates for two interacting systems in the plane. Probab. Th. Rel. Fields 73, 613–625 (1986). https://doi.org/10.1007/BF00324856
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00324856
Keywords
- Color
- Stochastic Process
- Probability Theory
- Large Time
- Mathematical Biology