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Quasi-nearest particle systems. (English) Zbl 0622.60116
In Bull. Am. Math. Soc. 83, 880-890 (1977; Zbl 0372.60149) F. Spitzer introduced and studied nearest particle systems. These are birth and death processes of single particle configurations in Z with the birth and death rates depending only on the distance to the nearest particle to the left and to the right of each site. T. M. Liggett [Ann. Probab. 11, 16-33 (1983; Zbl 0508.60081)] derived their ergodic behaviour in the attractive case.
In the present paper the authors study a generalization of these processes, which they call quasi nearest particle systems. Here the rates additionally depend on the nearest empty site behind the nearest particle to the left and to the right. They study reversibility and in the attractive case the ergodic behaviour and critical phenomena.
Reviewer: M.Mürmann
MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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[1] Spitzer, F., Stochastic time evolution of one dimensional infinite particle systems,Bull. Amer. Math. Soc.,83 (1977), 880–890. · Zbl 0372.60149 · doi:10.1090/S0002-9904-1977-14322-X
[2] Griffeath, D. and Liggett, T.M., Critical phenomena for Spitzer’s rebersible nearest particle systems,Ann. Probability,10 (1982), 881–895. · Zbl 0498.60090 · doi:10.1214/aop/1176993711
[3] Liggett, T.M., The stochastic evolution of infinite systems of interacting particles,Lect. Notes in Math.,598 (1977), 188–248. · Zbl 0363.60109
[4] Liggett, T.M., Attractive nearest particle systems,Ann. Probability,11 (1983), 16–33. · Zbl 0508.60081 · doi:10.1214/aop/1176993656
[5] Liggett, T.M., Two critical exponents for finite reversible nearest particle systems,Ann. Probability,11 (1983), 714–725. · Zbl 0527.60093 · doi:10.1214/aop/1176993516
[6] Gray, L., Controlled spin-flip systems,Ann. Probability,6 (1978) 953–974. · Zbl 0392.60084 · doi:10.1214/aop/1176995386
[7] Liu Xijian, A class of birth and death systems on Z., (1984), (to appear). · Zbl 0645.60107
[8] Dai Yonglong, Gibbs states and reverse random field.KEXUE TONGBAO 29 (1984) 1288–1291. · Zbl 0578.60097
[9] Tang Shouzheng. Reversibility of spin-flip processes,Acta. Math. Sinica,25 (1982) 306–314.
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