## The survival of nonattractive interacting particle systems on $$Z$$.(English)Zbl 1023.60087

Summary: We consider interacting particle systems (IPS) on $$Z$$ which allow five types of pairwise interaction: annihilation, birth, coalescence, death and exclusion with corresponding rates $$a,b,c,d,e$$. We show that whatever the values of $$a,c,d,e$$, if the birthrate is high enough, there is a positive probability the particle system will survive starting from any finite occupied set. In particular: an IPS with rates $$a,b,c,d,e$$ has a positive probability of survival if $$b>4d+6a$$, $$c+a\geq d+e$$, or $$b>7d+3a-3c+3e$$, $$c+a<d+e$$. We create a suitable supermartingale by extending the method used by R. Holley and T. M. Liggett [ibid. 6, 198-206 (1978; Zbl 0375.60111)] in their treatment of the contact process.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics

Zbl 0375.60111
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### References:

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