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Hunting submartingales in the jumping voter model and the biased annihilating branching process

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Aidan Sudbury
Aidan Sudbury*
Affiliation:
Monash University
*
∗ Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia. Email address: [email protected]
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Abstract

Two species (designated by 0's and 1's) compete for territory on a lattice according to the rules of a voter model, except that the 0's jump d0 spaces and the 1's jump d1 spaces. When d0 = d1 = 1 the model is the usual voter model. It is shown that in one dimension, if d1 > d0 and d0 = 1,2 and initially there are infinitely many blocks of 1's of length ≥ d1, then the 1's eliminate the 0's. It is believed this may be true whenever d1 > d0. In the biased annihilating branching process particles place offspring on empty neighbouring sites at rate λ and neighbouring pairs of particles coalesce at rate 1. In one dimension it is known to converge to the product measure density λ/(1+λ) when λ ≥ 1/3, and the initial configuration is non-zero and finite. This result is extended to λ ≥ 0.0347. Bounds on the edge-speed are given.

Keywords

Interacting particle systemsvoter modelsubmartingale

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 839 - 854
Copyright
Copyright © Applied Probability Trust 1999 

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References

Clifford, P. and Sudbury, A. W. (1973). A model for spatial conflict. Biometrika 60, 581–588.CrossRefGoogle Scholar
Griffeath, D. (1975). Ergodic theorems for graph interactions. Adv. Appl. Prob. 7, 179–194.CrossRefGoogle Scholar
Harris, T. E. (1976). On a class of set-valued Markov processes. Ann. Prob. 4, 175–194.Google Scholar
Holley, R. and Liggett, T. M. (1975). Ergodic theorems for weakly interacting systems and the voter model. Ann. Prob. 3, 643–663.CrossRefGoogle Scholar
Mountford, T. (1993). A coupling of finite particle systems. J. Appl. Prob. 30, 258–262.Google Scholar
Neuhauser, C. and Sudbury, A. W. (1993). The biased annihilating branching process. Adv. Appl. Prob. 25, 24–38.Google Scholar
Richardson, D. (1973). Random growth in a tesselation. Proc. Camb. Phil. Soc. 74, 515–528.Google Scholar
Sudbury, A. W. (1990). The branching annihilating process: an interacting particle system. Ann. Prob. 18, 581–601.Google Scholar
Sudbury, A. W. (1997). The convergence of the biased annihilating branching process and the double-flipping process in Zd . Stoch. Proc. Appl. 68, 255–264.CrossRefGoogle Scholar
Williams, T. and Bjerknes, R. (1972). Stochastic model for abnormal clone spread through epithelial base layer. Nature 236, 19–21.Google Scholar
Ziezold, H. and Grillenberger, C. (1998). On the critical infection rate of the one-dimensional basic contact process: numerical results. J. Appl. Prob. 25, 1–15.CrossRefGoogle Scholar