Reentrant phase diagram of branching annihilating random walks with one and two offspring

We investigate the phase diagram of branching annihilating random walks with one and two offspring in one dimension. A walker can hop to a nearest neighbor site or branch with one or two offspring with relative ratio. Two walkers annihilate immediately when they meet. In general, this model exhibits a continuous phase transition from an active state into the absorbing state (vacuum) at a finite hopping probability. We map out the phase diagram by Monte Carlo simulations that shows a reentrant phase transition from vacuum to an active state rand finally into vacuum again as the relative rate of the two-offspring branching process increases. This reentrant property apparently contradicts the conventional wisdom that increasing the number of offspring will tend to make the system more active. We show that the reentrant property is due to the static reflection symmetry of two-offspring branching processes and the conventional wisdom is recovered when the dynamic reflection symmetry is introduced instead of the static one.