Asymptotic Behavior of Densities for Two-Particle Annihilating Random Walks

Consider the system of particles on Z a 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 pA(t) and pB(t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densities pA(0)=pB(0) there is a change in behavior from d~<4, where p. l ( t )=pB(t )~C/ t d/4, to d>~4, where pA(t)= pB(t) ~ C/t as t ~ oo. For unequal initial densities pA(0)< pc(0), pA(t)~ e c ' / 7 in d = l , pA(t )~e -ct/l~ in d=2 , and pA(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.