Particles perform independent random walks on the integers, and are annihilated if they cross paths or land at the same point. The problem is to determine whether the origin is hit infinitely often. The answer is shown to depend on the initial distribution of particles in accordance with a "log log law." Several equivalent models are mentioned.
"Some Problems on Random Intervals and Annihilating Particles." Ann. Probab. 2 (5) 828 - 839, October, 1974. https://doi.org/10.1214/aop/1176996551