Asymptotic behavior of densities for two-particle annihilating random walks

Consider the system of particles onℤ^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 ρ_A(t) and ρ_B(t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densities ρ_A(0)=ρ_B(0) there is a change in behavior from d≤4, where ρ_A(t)=ρ_B(t)∼C/t^d/4, to d≥4, where ρ_A(t)=ρ_B(t)∼C/tas t→∞. For unequal initial densities ρ_A(0)<ρ_B(0), ρ_A(t)∼e^-c√l in d=1, ρ_A(t)∼e^{-Ct/log t} in d=2, and ρ_A(t)∼e^-Ct in d≥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.