On Pfaffian random point fields

TL;DR

This paper investigates Pfaffian random point fields using quaternion determinants, establishing conditions for validity, proving a CLT, and deriving Fredholm formulas with a novel quaternion extension of classical identities.

Contribution

It introduces a quaternion-based framework for Pfaffian point fields, including new conditions, a CLT, and Fredholm formulas, advancing the mathematical understanding of these fields.

Findings

Established sufficient conditions for valid quaternion kernel fields.

Proved a central limit theorem for Pfaffian point fields.

Derived Fredholm determinantal formulas using quaternion determinants.

Abstract

We study Pfaffian random point fields by using the Moore-Dyson quaternion determinants. First, we give sufficient conditions that ensure that a self-dual quaternion kernel defines a valid random point field, and then we prove a CLT for Pfaffian point fields. The proofs are based on a new quaternion extension of the Cauchy-Binet determinantal identity. In addition, we derive the Fredholm determinantal formulas for the Pfaffian point fields which use the quaternion determinant.