2000/03/21 by Maury Bramson, M. Bramson, Joel L. Lebowitz +3
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.math-ph/0003022
80 pages in an AMSTeX file, e-mail addresses: [email protected] and [email protected], replace for AMSTeX compilation error
openalex publication_date 2000/03/21 · arxiv created 2000/03/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the system of particles on \Bbb 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 [BrLe91a], 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 [BrLe99].