BKP hierarchy and Pfaffian point process

TL;DR

This paper establishes a connection between the BKP hierarchy and Pfaffian point processes, showing that correlation functions of shifted Schur measures can be expressed as Pfaffians and linking matrix integrals to Pfaffian processes.

Contribution

It introduces a novel relationship between BKP hierarchy and Pfaffian point processes, extending Okounkov's work on KP hierarchy and determinant processes.

Findings

Correlation functions are expressed as Pfaffians of skew-symmetric kernels.

Matrix integrals solving BKP hierarchy induce Pfaffian point processes.

The work generalizes determinant point process results to Pfaffian structures.

Abstract

Inspired by Okounkov's work [\emph{Selecta Mathematica}, 7(1):57--81, 2001] which relates KP hierarchy to determinant point process, we establish a relationship between BKP hierarchy and Pfaffian point process. We prove that the correlation function of the shifted Schur measures on strict partitions can be expressed as a Pfaffian of skew symmetric matrix kernel, whose elememts are certain vacuum expectations of neutral fermions. We further show that the matrix integrals solution of BKP hierarchy can also induce a certain Pfaffian point process.