Survival and extinction of caring double-branching annihilating random walk

Abstract

Branching annihilating random walk (BARW) is a generic term for a class of interacting particle systems on $\mathbb{Z}^d$ 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 $\mathbb{Z}$, and investigate its long-time 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.