Skip to main content Accessibility help

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

  • Aidan Sudbury (a1)


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 d 0 spaces and the 1's jump d 1 spaces. When d 0 = d 1 = 1 the model is the usual voter model. It is shown that in one dimension, if d 1 > d 0 and d 0 = 1,2 and initially there are infinitely many blocks of 1's of length ≥ d 1, then the 1's eliminate the 0's. It is believed this may be true whenever d 1 > d 0. 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.


Corresponding author

Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia. Email address:


Hide All
Clifford, P. and Sudbury, A. W. (1973). A model for spatial conflict. Biometrika 60, 581588.
Griffeath, D. (1975). Ergodic theorems for graph interactions. Adv. Appl. Prob. 7, 179194.
Harris, T. E. (1976). On a class of set-valued Markov processes. Ann. Prob. 4, 175194.
Holley, R. and Liggett, T. M. (1975). Ergodic theorems for weakly interacting systems and the voter model. Ann. Prob. 3, 643663.
Mountford, T. (1993). A coupling of finite particle systems. J. Appl. Prob. 30, 258262.
Neuhauser, C. and Sudbury, A. W. (1993). The biased annihilating branching process. Adv. Appl. Prob. 25, 2438.
Richardson, D. (1973). Random growth in a tesselation. Proc. Camb. Phil. Soc. 74, 515528.
Sudbury, A. W. (1990). The branching annihilating process: an interacting particle system. Ann. Prob. 18, 581601.
Sudbury, A. W. (1997). The convergence of the biased annihilating branching process and the double-flipping process in Zd . Stoch. Proc. Appl. 68, 255264.
Williams, T. and Bjerknes, R. (1972). Stochastic model for abnormal clone spread through epithelial base layer. Nature 236, 1921.
Ziezold, H. and Grillenberger, C. (1998). On the critical infection rate of the one-dimensional basic contact process: numerical results. J. Appl. Prob. 25, 115.


MSC classification

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

  • Aidan Sudbury (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.