Phase transition in a double branching annihilating random walk

Abstract

This paper investigates the long-time behavior of double branching annihilating random walkers with nearest-neighbor dependent rates. The system consists of even number of particles which can execute nearest-neighbor random walk and they can as well give birth in a parity conserving manner to two other particles with rates 1 and b, respectively, until they meet. Upon meeting, each of the adjacent particles can branch with rate p • b while it can annihilate, i.e. hop on, the other particle with rate p for some 0 < p ≤ 1. This process first appeared in [2] and can be considered as the extension of that of [15]. We prove that in some region of the parameters (p, b), the process survives with positive probability. Combining the extinction result of [15] it shows a phase transition phenomenon for this model. In some sense our result also shows the sharpness of the assumptions of [1]. We use similar arguments that was developed by M. Bramson and L. Gray in [5].