Asymptotic behavior of the density for two-particle annihilating exclusion

We consider a stochastic process which presents an evolution of particles of two types,A andB, onZ^d with annihilations between particles of opposite types. Initially, at each site ofZ^d, independently of the other sites, we put a particle with probability 2<1 and assign to it one of two types with equal chances. Each particle evolves onZ^d in the following manner: independently from the others, it waits an exponential time with mean 1, chooses one of its neighboring sites on the latticeZ^d with equal probabilities, and jumps to the site chosen. If the site to which a particle attempts to move is occupied by another particle of the same type, the jump is suppressed; if it is occupied by a particle of the opposite type, then both are annihilated and disappear from the system. The considered process may serve as a model for the chemical reactionA+Binert. Let (t) denote the density of particles in this process at timet. We prove that there exist absolute finite constantsc(d) andC(d) such that for all sufficiently larget,c(d)t^–d/4(t)C(d)t^–d/4 in the dimensionsd4 andc(d)t^–1(t)C(d)t^–1 in all higher dimensions. This completes and makes more precise the results obtained by us earlier and shows that asymptotically the density behaves like that in a similar process called two-particle annihilating random walks which was studied by Bramson and Lebowitz. Our proofs are based on the approach developed in their and our works. We use the basic properties of random walk and various tools which have been designed to study simple symmetric exclusion processes.