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Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  22 February 2016

Daniela Bertacchi ,
Fábio Prates Machado  and
Fabio Zucca
Daniela Bertacchi*
Affiliation:
Università di Milano-Bicocca
Fábio Prates Machado*
Affiliation:
Universitade de São Paulo
Fabio Zucca*
Affiliation:
Politecnico di Milano
*
∗ Postal address: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy. Email address: [email protected]
∗∗ Postal address: Instituto de Matemática e Estatística, Universitade de São Paulo, Rua do Matão 1010, São Paulo, Brasil. Email address: [email protected]
∗∗∗ Postal address: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Email address: [email protected]
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Abstract

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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).

Keywords

Inhomogeneous random walkfrog modelegg modellocal survivalglobal survival

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 256 - 278
Copyright
© Applied Probability Trust 

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