Propagation and extinction in branching annihilating random walks.

We investigate the temporal evolution and spatial propagation of branching annihilating random walks in one dimension. Depending on the branching and annihilation rates, a few-particle initial state can evolve to a propagating finite density wave, or extinction may occur, in which the number of particles vanishes in the long-time limit. The number parity conserving case where 2-offspring are produced in each branching event can be solved exactly for unit reaction probability, from which qualitative features of the transition between propagation and extinction, as well as intriguing parity-specific effects are elucidated. An approximate analysis is developed to treat this transition for general BAW processes. A scaling description suggests that the critical exponents which describe the vanishing of the particle density at the transition are unrelated to those of conventional models, such as Reggeon Field Theory. In the branching annihilating random walk (BAW), a single random walk branches at some specified rate and two random walkers annihilate at another rate when they meet. 1−4 As a function of these rates, the number of random walkers may grow without bound, reach a finite limiting number, or vanish asymptotically. Our goal, in this paper, is to determine some of the long-time properties of this BAW process. We are particularly interested in understanding the kinetics and density distribution when the initial state consists of a small number of localized particles. Interest in this process has several motivations. First, considerable theoretical effort has been devoted to establishing the existence of and quantifying the non-equilibrium phase transition between " propagation " and " extinction " for a variety of interacting particle systems 5 which are closely related to BAWs. Here the term extinction refers to the situation where annihilation dominates over branching and an initially localized population of particles ultimately disappears. In the complementary case, branching dominates over annihilation and an initially localized population evolves into a propagating wavefront which advances into the otherwise empty system. Typical examples of these phenomena 1