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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

Citations:

Zbl 0375.60111
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References:

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