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

We derive a self-duality relation for a one-dimensional model of branching and annihilating random walkers with an even number of offspring on nearest-neighbour sites. With the duality relation and by deriving further 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. In the absence of particle diffusion the branching process (with infinite annihilation rate) is strongly non-ergodic.