The branching annihilating process: An interacting particle system
The branching annihilating process is an interacting particle system on \({\mathbb{Z}}^ d\) with at most one particle at every site. Each particle places an offspring at rate 1 on one of the neighbouring sites and when this site is occupied, both particles annihilate each other. This is a special case of the branching annihilating walk introduced by \textit{M. Bramson} and \textit{L. Gray} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 68, 447-460 (1985; Zbl 0537.60099)], where in addition particles jump at rate \(\rho\) to a neighbouring site. They however only study the one- dimensional case. In the present paper it is shown, that the branching annihilating process, starting with a non-empty set of occupied sites, converges to the symmetric product measure. The author also treats the double branching annihilating process in one dimension, where a particle tries to place offsprings on both neighbouring sites. For a finite number of particles the limiting behaviour depends on whether the initial number is odd or even. The symmetric product measure is not the only limiting measure. The proof of these results uses the fact, that the border of an exclusion process is a double branching annihilating process.
scientific article; zbMATH DE number 4157652