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.


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


Zbl 0375.60111
Full Text: DOI


[1] Bezuidenhout, C. and Grimmett, G. (1990). The critical contact process dies out. Ann.Probab. 18 1462-1482. · Zbl 0718.60109
[2] Bramson, M. and Gray, L. (1985). The survival of the branching annihilating random walk.Wahrsch.Verw.Gebiete 68 447-460. · Zbl 0537.60099
[3] Durrett, R. and Neuhauser, C. (1994). Particle systems and reaction-diffusion equations. Ann. Probab. 22 289-333. · Zbl 0799.60093
[4] Holley, R. and Liggett, T. M. (1978). The survival of contact processes. Ann.Probab.6 198-206. · Zbl 0375.60111
[5] Konno, N. (1994). Phase Transitions of Interacting Particle Systems. World Scientific, Singapore. · Zbl 0847.60088
[6] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York. Liggett, T. M. (1995a). Improved upper bounds for the contact process critical value. Ann. Probab. 23 697-723. Liggett, T. M. (1995b). Survival of discrete time growth models with applications to oriented percolation. Ann.Appl.Probab.5 613-636. · Zbl 0832.60093
[7] Sudbury, A. W. (1990). The branching annihilating process: an interacting particle system. Ann. Probab. 18 581-601. · Zbl 0705.60094
[8] Sudbury, A. W. (1998). A method for finding bounds on critical values for nonattractive interacting particle systems. J.Phys.A 31 8323-8331. · Zbl 0979.82022
[9] Sudbury, A. W. (2000). Dual families of interacting particle systems on graphs. J.Theoret.Probab. 13 3. · Zbl 0968.60096
[10] Ziezold, H. and Grillenberger, C. (1998). On the critical infection rate of the one-dimensional basic contact process: numerical results. J.Appl.Probab.25 1-15. JSTOR: · Zbl 0643.60095
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