Persistence in the one-dimensional A+B -> 0 reaction-diffusion model - art. no. 041105

The persistence properties of a set of random walkers obeying the A + B --> empty set reaction, with equal initial density of particles and homogeneou s initial conditions, is studied using two definitions of persistence. The probability P(t) that an annihilation process has not occurred at a given s ite has the asymptotic form P(t) similar to const+ t(-theta), where theta i s the persistence exponent (type I persistence). We argue that, for a densi ty of particles rho >> 1, this nontrivial exponent is identical to that gov erning the persistence properties of the one-dimensional diffusion equation , partial derivative (t)phi=partial derivative (xx)phi, where theta similar or equal to 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, rho (o) << 1, we find theta similar or equal to 1/4 asymptotically. The pr obability that a site remains unvisited by any random walker (type II persi stence) is also investigated and found to decay with a stretched exponentia l form, P(t)similar to exp(-const x rho (1/2)(o) t(1/4)), provided rho (o)< <1. A heuristic argument for this behavior, based on an exactly solvable to y model, is presented.