Exponents appearing in heterogeneous reaction-diffusion models in one dimension

We study the following one-dimensional (1D) two-species reaction-diffusion model: there is a small concentration of B particles with diffusion constant D-B in an homogenous background of W particles with diffusion constant D-W; two W particles of the majority species either coagulate (W+W-->W) or annihilate (W+W-->0) with the respective probabilities p(c)=(q-2)/(q-1) and p(a)=1/(q-1); a B particle and a W particle annihilate (W+B-->0) with probability 1. The exponent theta(q,lambda=D-g/D-W) describing the asymptotic time decay of the minority B species concentration can be viewed as a generalization of the exponent of persistent spins in the zero-temperature Glauber dynamics of the 1D q-state Potts model starting from a random initial condition: the W particles represent domain walls, and the exponent theta(q,lambda) characterizes the time decay of the probability that a diffusive ''spectator'' does not meet a domain wall up to time t. We extend the methods introduced by Derrida, Hakim, and Pasquier [Phys. Rev. Lett. 75, 751 (1995); J. Stat. Phys. (to be published)] for the problem of persistent spins, to compute the exponent theta(q,lambda) in perturbation at first order in (q-1) for arbitrary lambda and at first order in lambda for arbitrary q.