Bertacchi, Daniela; Machado, Fábio Prates; Zucca, Fabio Local and global survival for nonhomogeneous random walk systems on \(\mathbb Z\). (English) Zbl 1302.60131 Adv. Appl. Probab. 46, No. 1, 256-278 (2014). The authors study a system of interacting particles on \(\mathbb Z\), where at time 0 there is one active particle at 0 and one inactive particle at each vertex of positive integers. Particles become active if an active particle jumps to their location. The particle which at time 0 is at \(n\), once activated, has a geometrically distributed lifespan with parameter depending on \(n\) and, while alive, performs a nearest neighbor random walk with probability depending on \(n\), so the system is an inhomogeneous process of random walkers. This system can be seen as a model for disease or information spreading. It is said that there is local survival if the event that site 0 is visited infinitely many times has positive probability. It is said that there is global survival if, with positive probability, at any time there is at least one active particle.The authors study local and global survival of an inhomogeneous system of interacting particles where the initial configuration is given by one particle per site with geometric lifespans. They give conditions for global and local survival, and infinite activation, both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan. In particular, in the immortal case, a 0-1 law for the probability of local survival when all particles drift to the right is proved. In addition, the authors give sufficient conditions for local survival or local extinction when all particles drift to the left. Reviewer: Anatoliy Pogorui (Zhytomyr) Cited in 6 Documents MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G50 Sums of independent random variables; random walks Keywords:inhomogeneous random walk; local survival; global survival; frog model; egg model × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). The shape theorem for the frog model. Ann. Appl. Prob. 12 , 533-546. · Zbl 1013.60081 · doi:10.1214/aoap/1026915614 [2] Alves, O. S. M., Machado, F. P., Popov, S. Yu. and Ravishankar, K. (2001). The shape theorem for the frog model with random initial configuration. Markov Process. Relat. Fields 7 , 525-539. · Zbl 0991.60097 [3] Bertacchi, D. and Zucca, F. (2008). Critical behaviors and critical values of branching random walks on multigraphs. J. Appl. Prob. 45 , 481-497. · Zbl 1144.60057 · doi:10.1239/jap/1214950362 [4] Bertacchi, D. and Zucca, F. 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