Persistence in the One-Dimensional A+B -> 0 Reaction-Diffusion Model

Abstract

The persistence properties of a set of random walkers obeying the A+B -> 0 reaction, with equal initial density of particles and homogeneous initial conditions, is studied using two definitions of persistence. The probability, P(t), that an annihilation process has not occurred at a given site has the asymptotic form $P(t) -> const + t^{-θ}$, where $θ$ is the persistence exponent (``type I persistence''). We argue that, for a density of particles $ρ>> 1$, this non-trivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, where $θ\approx 0.1207$. In the case of an initially low density, $ρ_0 << 1$, we find $θ\approx 1/4$ asymptotically. The probability that a site remains unvisited by any random walker (``type II persistence'') is also investigated and found to decay with a stretched exponential form, $P(t) \sim \exp(-const ρ_0^{1/2}t^{1/4})$, provided $ρ_0 << 1$. A heuristic argument for this behavior, based on an exactly solvable toy model, is presented.