Local and global survival for nonhomogeneous random walk systems on Z

TL;DR

This paper analyzes the conditions for survival and activation in a nonhomogeneous interacting random walk system on Z, considering both immortal and mortal particles with various drift behaviors and providing explicit phase diagrams for specific cases.

Contribution

It offers new criteria for global and local survival in nonhomogeneous random walks with activation dynamics, including cases with mortal particles and explicit phase diagrams.

Findings

0-1 law for local survival with all particles drifting right

Conditions for local survival or extinction with particles drifting left

Complete phase diagrams for specific asymptotic behaviors of parameters

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 $n \ge 1$. 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 $l_n$. 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 $p_n \in [0, 1]$. 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 $1/2-l_n \sim \pm 1/n^\alpha$, $p_n = 1$ and $1_n-1/2 \sim \pm 1/n^\alpha$, $1-p_n \sim 1/n^\beta$ (where $\alpha,\beta > 0$).