Random walks systems with finite lifetime on $ \bbZ $

TL;DR

This paper studies a non-homogeneous random walk system on the integer line where particles activate others upon encounter, providing conditions for the process to sustain indefinitely.

Contribution

It establishes necessary and sufficient conditions for the survival of the particle activation process in a finite lifetime random walk system.

Findings

Derived criteria for process survival

Characterized conditions for infinite activation

Analyzed the impact of jump limits on survival

Abstract

We consider a non-homogeneous random walks system on $\bbZ$ in which each active particle performs a nearest neighbor random walk and activates all inactive particles it encounters up to a total amount of $L$ jumps. We present necessary and sufficient conditions for the process to survive, which means that an infinite number of random walks become activated.