Examples of Interacting Particle Systems on $$\mathbb {Z}$$ as Pfaffian Point Processes: Coalescing–Branching Random Walks and Annihilating Random Walks with Immigration

Two classes of interacting particle systems on $$\mathbb {Z}$$Z are shown to be Pfaffian point processes, at any fixed time and for all deterministic initial conditions. The first comprises coalescing and branching random walks, the second annihilating random walks with pairwise immigration. Various limiting Pfaffian point processes on $$\mathbb {R}$$R are found by diffusive rescaling, including the point set process for the Brownian web and Brownian net.