Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle   Reactions
  Consider the system of particles on ${\Bbb Z}^d$ 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 $A+B\to inert$. In [BrLe91a], 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 $d\geq 4$, as $t\to\infty$. 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 $d\geq 4$ will be handled in a future work [BrLe99]. 
M. Bramson, and J. L. Lebowitz