TY - JOUR

T1 - Spatial structure in low dimensions for diffusion limited two-particle reactions

AU - Bramson, Maury

AU - Lebowitz, Joel L.

PY - 2001/2

Y1 - 2001/2

N2 - 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 A + B → inert. 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.

AB - 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 A + B → inert. 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.

KW - Annihilating random walks

KW - Asymptotic densities

KW - Diffusion limited reaction

KW - Spatial structure

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U2 - 10.1214/aoap/998926989

DO - 10.1214/aoap/998926989

M3 - Article

AN - SCOPUS:0035592702

SN - 1050-5164

VL - 11

SP - 121

EP - 181

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 1

ER -