Abstract | 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 $n ge1$. Particles
become active when hit by another active particle. Once activated
they 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. |