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Preprint 5-2011

On Survival and Extinction of Caring Double-Branching Annihilating Random Walk

Source file is available as :   Portable Document Format (PDF)

Author(s) : Jochen Blath , Noemi Kurt

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 5-2011

MSC 2000

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes
60J27 Markov chains with continuous parameter

Abstract :
Branching annihilating random walk (BARW) is a generic term for a class of interacting particle systems on Zd in which, as time evolves, particles execute random walks, produce offspring (on neighbouring sites) and (instantaneously) disappear when they meet other particles. Much of the interest in such models stems from the fact that they typically lack a monotonicity property called attractiveness, which in general makes them exceptionally hard to analyse and in particular highly sensitive in their qualitative long-time behaviour to even slight alterations of the branching and annihilation mechanisms. In this short note, we introduce so-called caring double-branching annihilating random walk (cDBARW) on Z, and investigate its longtime behaviour. It turns out that it either allows survival with positive probability if the branching rate is greater than 1/2, or a.s. extinction if the branching rate is smaller than 1/3 and (additionally) branchings are only admitted for particles which have at least one neighbouring particle (so-called �cooperative branching�). Further, we show a.s. extinction for all branching rates for a variant of this model, where branching is only allowed if offspring can be placed at odd distance between each other. It is the latter (extinction-type) results which seem remarkable, since they appear to hint at a general extinction result for a non-trivial parameter range in the so-called �parity-preserving universality class�, suggesting the existence of a �true� phase transition. The rigorous proof of such a non-trivial phase transition remains a particularly challenging open problem.

Keywords : Branching Annihilating Random Walk, extinction, survival, interface duality, swapping voter model

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