Spatial structure in low dimensions for diffusion limited two-particle reactions

Consider the system of particles on Z(d) where particles are of two types, A and B, and execute simple random walks in continuous time. Particles do not interact with their own type, but when a type A particle meets a type B particle, both disappear. Initially, particles are assumed to be distributed according to homogeneous Poisson random fields, with equal intensities for the two types. This system serves as a model for the chemical reaction A + B --> inert. In Bramson and Lebowitz [7], the densities of the two types of particles were shown to decay asymptotically like 1/t(d/4) for d < 4 and 1/t for d greater than or equal to 4, as t --> infinity. This change in behavior from low to high dimensions corresponds to a change in spatial structure. In d < 4, particle types segregate, with only one type present locally. After suitable rescaling, the process converges to a limit, with density given by a Gaussian process. In d > 4, both particle types are, at large times, present locally in concentrations not depending on the type, location or realization. In d = 4, both particle types are present locally, but with varying concentrations. Here, we analyze this behavior in d < 4; the behavior for d greater than or equal to 4 will be handled in a future work by the authors.