The spread of a rumor or infection in a moving population. (English) Zbl 1111.60074

The authors study a particle system with two types of particles: \(A\)-particles perform independently continuous time simple random walk with rate \(D_A\). \(B\)-particles perform simple random walk with rate \(D_B\). Initally there is a Poisson process of \(A\)-particles and finitely many \(B\)-particles. \(A\)- and \(B\)-particles interact in that if an \(A\)-particle meets a \(B\)-particle, the former is transformed into a \(B\)-particle instantly. \(B\)-particles are considered to model a rumor or an infection. The authors show that if the diffusion constants coincide, the \(B\)-particles invade by time \(t\) cubes of sidelength of order \(t\). A slightly weaker result is shown in the general case.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
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