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

Maury Bramson, Joel L. Lebowitz

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)121-181
Number of pages61
JournalAnnals of Applied Probability
Issue number1
StatePublished - Feb 2001


  • Annihilating random walks
  • Asymptotic densities
  • Diffusion limited reaction
  • Spatial structure


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