M. Bramson (bramson@math.umn.edu) and J. L. Lebowitz, (lebowitz@sakharov.rutgers.edu)
Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle
Reactions
(1306K, ps file)
ABSTRACT. 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].