Consider the system of particles on Zd 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 , the densities of the two types of particles were shown to decay asymptotically like 1/td/4 for d<4 and 1/t for d > 4, as t → ∞. 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=4 will be handled in a future work by the authors.
"Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle Reactions." Ann. Appl. Probab. 11 (1) 121 - 181, February 2001. https://doi.org/10.1214/aoap/998926989