Pfaffian formulae for one dimensional coalescing and annihilating systems

TL;DR

This paper derives Pfaffian formulas for the distribution of one-dimensional coalescing and annihilating Brownian particles, revealing connections to random matrix eigenvalues and providing exact asymptotics for particle densities.

Contribution

It introduces Pfaffian point process representations for these particle systems under maximal entrance laws, linking them to real Ginibre ensemble eigenvalues and deriving asymptotic density results.

Findings

Distribution of particles is Pfaffian under maximal entrance laws

Connections established between particle systems and real Ginibre ensemble

Exact large-time asymptotics for n-point density functions

Abstract

The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at a fixed time is shown to be Pfaffian point processes closely related to the Pfaffian point process describing one dimensional distribution of real eigenvalues in the real Ginibre ensemble of random matrices. As an application, an exact large time asymptotic for the n-point density function for coalescing particles is derived.