ASYMPTOTIC-BEHAVIOR OF DENSITIES FOR 2-PARTICLE ANNIHILATING RANDOM-WALKS

Consider the system of particles on Z(d) 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 an A-particle meets a B-particle, both disappear, i.e., are annihilated. This system serves as a model for the chemical reaction A + B --> inert. We analyze the limiting behavior of the densities rho A(t) and rho B(t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densities rho A(0) = rho B(0) there is a change in behavior from d less-than-or-equal-to 4, where rho A(t) = rho B(t) approximately C/t(d/4), to d greater-than-or-equal-to 4, where rho A(t) = rho B(t) approximately C/t as t --> infinity. For unequal initial densities rho A(0) < rho B(0), rho A(t) approximately e-c square-root t in d = 1, rho A(t) approximately e-Ct/log t in d = 2, and rho A(t) approximately e-Ct in d greater-than-or-equal-to 3. The term C depends on the initial densities and changes with d. Techniques are from interacting particle systems. The behavior for this two-particle annihilation process has similarities to those for coalescing random walks (A + A --> A) and annihilating random walks (A + A --> inert). The analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system.