Persistence in the one-dimensional A+B⃗∅ reaction-diffusion model

The persistence properties of a set of random walkers obeying the A+B⃗∅ 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 nontrivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, ∂_tφ=∂_xxφ, where θ≃0.1207 [S. N. Majumdar, C. Sire, A. J. Bray, and S. J. Cornell, Phys. Rev. Lett. 77, 2867 (1996)]. In the case of an initial low density, ρ₀≪1, we find θ≃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)∼exp(-const×ρ₀^1/2t^1/4), provided ρ₀≪1. A heuristic argument for this behavior, based on an exactly solvable toy model, is presented.