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February 2001 Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle Reactions
Maury Bramson, Joel L. Lebowitz
Ann. Appl. Probab. 11(1): 121-181 (February 2001). DOI: 10.1214/aoap/998926989


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 ABinert. In Bramson and Lebowitz [7], 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.


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Maury Bramson. Joel L. Lebowitz. "Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle Reactions." Ann. Appl. Probab. 11 (1) 121 - 181, February 2001.


Published: February 2001
First available in Project Euclid: 27 August 2001

zbMATH: 1016.60088
MathSciNet: MR1825462
Digital Object Identifier: 10.1214/aoap/998926989

Primary: 60K35

Keywords: annihilating random walks , asymptotic densities , diffusion limited reaction , spatial structure

Rights: Copyright © 2001 Institute of Mathematical Statistics

Vol.11 • No. 1 • February 2001
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