Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: coalescing branching random walks and annihilating random walks with immigration

TL;DR

This paper demonstrates that certain interacting particle systems on the integer lattice are Pfaffian point processes at fixed times, providing explicit formulas and limits for systems like coalescing, branching, and annihilating random walks with immigration.

Contribution

It establishes the Pfaffian structure for these particle systems and derives their limiting processes under diffusive scaling, including the Brownian net.

Findings

Coalescing and branching random walks are Pfaffian point processes.

Annihilating random walks with immigration are Pfaffian point processes.

Limiting processes include the Brownian net and other diffusive limits.

Abstract

Two classes of interacting particle systems on $\mathbb{Z}$ are shown to be Pfaffian point processes at fixed times, 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 $\mathbb{R}$ are found by diffusive rescaling, including the point set process for the Brownian net.