Asymptotic behavior of densities for two-particle annihilating random walks

Abstract

Consider the system of particles onℤ d where particles are of two types—A andB—and execute simple random walks in continuous time. Particles do not interact with their own type, but when anA-particle meets aB-particle, both disappear, i.e., are annihilated. This system serves as a model for the chemical reactionA+B→ inert. We analyze the limiting behavior of the densitiesρ A (t) andρ B (t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densitiesρ A (0)=ρ B (0) there is a change in behavior fromd⩽4, whereρ A (t)=ρ B (t)∼C/t d /4, tod⩾4, whereρ A (t)=ρ B (t)∼C/tast→∞. For unequal initial densitiesρ A (0)<ρ B (0),ρ A (t)∼e−c√l ind=1,ρ A (t)∼e−Ct/logt ind=2, andρ A (t)∼e−Ct ind⩾3. The termC depends on the initial densities and changes withd. 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.