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

TL;DR

This paper demonstrates that certain interacting particle systems on the integer lattice, involving annihilation and coalescence, form Pfaffian point processes, and derives explicit kernels for their diffusion limits, revealing connections to random matrix theory.

Contribution

It establishes the Pfaffian structure for a broad class of annihilating and coalescing particle systems and derives explicit kernels for their diffusion limits, linking to random matrix ensembles.

Findings

Pfaffian point process structure for particle systems on $ Z$

Explicit kernels for coalescing and annihilating Brownian motions

Connections to eigenvalue distributions in random matrix theory

Abstract

A class of interacting particle systems on $\mathbb{Z}$, involving instantaneously annihilating or coalescing nearest neighbour random walks, are shown to be Pfaffan point processes for all deterministic initial conditions. As diffusion limits, explicit Pfaffan kernels are derived for a variety of coalescing and annihilating Brownian systems. For Brownian motions on $\mathbb{R}$, depending on the initial conditions, the corresponding kernels are closely related to the bulk and edge scaling limits of the Pfaffan point process for real eigenvalues for the real Ginibre ensemble of random matrices. For Brownian motions on $\mathbb{R}_{+}$ with absorbing or reflected boundary conditions at zero new interesting Pfaffan kernels appear. We illustrate the utility of the Pfaffan structure by determining the extreme statistics of the rightmost particle for the purely annihilating Brownian motions, and also computing the probability of overcrowded regions for all models.