Local and global survival for nonhomogeneous random walk systems on Z

This paper studies an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site n ≥ 1. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left-jump probability l... This paper gives conditions for global survival, local survival, and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). More precisely, once activated, the particle at n survives at each step with probability p... ... [0, 1]. In particular, in the immortal case, it proves a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, this paper gives sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, it provides sufficient conditions for global survival, local survival, and local extinction (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: this paper describes completely the phase diagram in the cases 1/2 - l... ~ ± 1 / n...,p... = 1 and 1/2 - l... ~ ± 1 / n...,1 - p... ~ 1 / n...(where α, β > 0). (ProQuest: ... denotes formulae/symbols omitted.)