# zbMATH — the first resource for mathematics

Survival and coexistence in interacting particle systems. (English) Zbl 0832.60094
Grimmett, Geoffrey (ed.), Probability and phase transition. Proceedings of the NATO Advanced Study Institute on probability theory of spatial disorder and phase transition, Cambridge, UK, July 4-16, 1993. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 420, 209-226 (1994).
This paper reports a new technique discovered recently by the author. That is, using some carefully designed perturbations of the renewal measure instead of the original renewal measure. Two applications of the technique are reported here. First, the upper bound $$\lambda_c$$ of the basic one-dimensional contact process is improved from 2 to 1.942. Secondly, a conjecture made by J. T. Cox and R. Durrett [in: Radom walks, Brownian motion, and interacting particle systems. Prog. Probab. 28, 189-201 (1991)] for the threshold voter model is proved by the author. Finally, about 300 publications on interacting particle systems after the author’s book appeared [“Interacting particle systems” (1985; Zbl 0559.60078)] are collected at the end of the paper.
For the entire collection see [Zbl 0818.00015].

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory