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

#### 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 ≥ 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 ln. 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). More precisely, once activated, the particle at n survives at each step with probability pn ∈ [0, 1]. 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 (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: we describe completely the phase diagram in the cases ½ - ln ~ ± 1 / nα, pn = 1 and ½ - ln ~ ± 1 / nα, 1 - pn ~ 1 / nβ (where α, β > 0).

#### Article information

Source
Adv. in Appl. Probab., Volume 46, Number 1 (2014), 256-278.

Dates
First available in Project Euclid: 1 April 2014

https://projecteuclid.org/euclid.aap/1396360113

Digital Object Identifier
doi:10.1239/aap/1396360113

Mathematical Reviews number (MathSciNet)
MR3189058

Zentralblatt MATH identifier
1302.60131

#### Citation

Bertacchi, Daniela; Machado, Fábio Prates; Zucca, Fabio. Local and global survival for nonhomogeneous random walk systems on Z. Adv. in Appl. Probab. 46 (2014), no. 1, 256--278. doi:10.1239/aap/1396360113. https://projecteuclid.org/euclid.aap/1396360113

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