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

TL;DR

This paper investigates the spatial structure and asymptotic behavior of a two-particle reaction system on a lattice in dimensions less than four, revealing segregation and Gaussian limits in low dimensions.

Contribution

It provides a detailed analysis of the spatial structure and limiting behavior of the reaction system for dimensions less than four, complementing previous work on higher dimensions.

Findings

Particles segregate in dimensions less than four.

Rescaled processes converge to Gaussian limits.

Density decay follows specific power laws depending on dimension.

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].