The annihilating process
In this particle system the only interaction is that each particle kills a neighboring particle at a fixed rate. There is no birth and no movement, hence an isolated particle remains for ever. The authors show that when the initial configuration is the set of integers, the distribution of particles at all times is a renewal process and that the probability that a site remains occupied for all times converges to \(1/e\). Using a Harris graphical construction they describe explicitly the distribution of the renewal process for all times. They extend some of the results to a tree and leave open the problem in dimensions \(d\geq 2\).
scientific article; zbMATH DE number 1625619