The Annals of Applied Probability

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

Maury Bramson and Joel L. Lebowitz

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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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Ann. Appl. Probab., Volume 11, Number 1 (2001), 121-181.

First available in Project Euclid: 27 August 2001

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

diffusion limited reaction annihilating random walks asymptotic densities spatial structure


Bramson, Maury; Lebowitz, Joel L. Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle Reactions. Ann. Appl. Probab. 11 (2001), no. 1, 121--181. doi:10.1214/aoap/998926989.

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