Branching-annihilating random walks in one dimension: Some exact results

TL;DR

This paper derives exact results for a one-dimensional branching-annihilating random walk model, revealing the nature of phase transitions and fluctuation behaviors, and clarifying open problems in the field.

Contribution

It introduces a self-duality relation for the model and provides exact results in certain limits, enhancing understanding of phase transition properties.

Findings

Transition is mean-field-like in certain limits

Fluctuations deviate from mean-field on the active side

Finite systems approach absorbing state very slowly

Abstract

We derive a self-duality relation for a one-dimensional model of branching and annihilating random walkers with an even number of offsprings. With the duality relation and by deriving exact results in some limiting cases involving fast diffusion we obtain new information on the location and nature of the phase transition line between an active stationary state (non-zero density) and an absorbing state (extinction of all particles), thus clarifying some so far open problems. In these limits the transition is mean-field-like, but on the active side of the phase transition line the fluctuation in the number of particles deviates from its mean-field value. We also show that well within the active region of the phase diagram a finite system approaches the absorbing state very slowly on a time scale which diverges exponentially in system size.