Local and global survival for nonhomogeneous random walk systems on Z

Details Local and global survival for nonhomogeneous random walk systems on Z Local and global survival for nonhomogeneous random walk systems on Z

2012-01-16

arxiv.org/abs/1201.3324v2

Mathematics

Probability

20 pages, 2 figures, fixed some typos and shortened the introduction

Scientific paper

We study 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 $n \ge 1$. Particles become active when hit by another active particle. Once activated perform an asymmetric nearest neighbour random walk which depends only on the starting location of the particle. We give 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). In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival and local extinction. Analysis of explicit examples is provided.

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