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The complete convergence theorem for coexistent threshold voter models. (English) Zbl 0974.60093
The threshold voter model was introduced by {\it J. T. Cox} and {\it R. Durrett} [in: Random walks, Brownian motion, and interacting particle systems. Prog. Probab. 28, 189-201 (1991)] which conjectured that it converges to a stationary distribution. In the present paper one proves a generalization of this conjecture.

60K35Interacting random processes; statistical mechanics type models; percolation theory
60J65Brownian motion
Full Text: DOI
[1] Andjel, E. D., Liggett, T. M. and Mountford, T. (1992). Clustering in one-dimensional threshold voter models. Stochastic Process. Appl. 42 73-90. · Zbl 0752.60086 · doi:10.1016/0304-4149(92)90027-N
[2] Bezuidenhout, C. and Gray, L. (1994). Critical attractive spin systems. Ann. Probab. 22 1160- 1194. · Zbl 0819.60094 · doi:10.1214/aop/1176988599
[3] Bramson, M., Ding, W. D. and Durrett, R. (1991). Annihilating branching processes. Stochastic Process. Appl. 37 1-17. · Zbl 0745.60085 · doi:10.1016/0304-4149(91)90056-I
[4] Cox, J. T. and Durrett, R. (1991). Nonlinear voter models. In Random Walks, Brownian Motion and Interacting Particle Systems. A Festschrift in Honor of Frank Spitzer 189-201. Birkhäuser, Boston. · Zbl 0825.60053
[5] Durrett, R. (1995). Ten Lectures on Particle Systems. Lecture Notes in Math. 1608. Springer, New York. · Zbl 0840.60088
[6] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York. · Zbl 0559.60078
[7] Liggett, T. M. (1994). Coexistence in threshold voter models. Ann. Probab. 22 764-802. · Zbl 0814.60094 · doi:10.1214/aop/1176988729