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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.

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