## DIMACS TR: 2000-04

## Spatial Structure in Low Dimensions for
Diffusion Limited Two-Particle Reactions

### Authors: M. Bramson and J. L. Lebowitz

**
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].

Paper Available at:
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