On Fredholm Pfaffians and Riemann-Hilbert problems

TL;DR

This paper demonstrates how Fredholm Pfaffians associated with certain kernels can be computed using Riemann-Hilbert problems, enabling asymptotic analysis through this characterization.

Contribution

It introduces a method to compute Fredholm Pfaffians via Riemann-Hilbert problems for specific kernel classes, connecting Pfaffian evaluation with integrable systems.

Findings

Fredholm Pfaffians can be expressed through Riemann-Hilbert problems.

Asymptotic results follow naturally from the Riemann-Hilbert framework.

Applicable to kernels of additive Hankel and truncated Wiener-Hopf types.

Abstract

It is shown how classes of Fredholm Pfaffians can be computed in terms of canonical, auxiliary Riemann-Hilbert problems as soon as the main kernel in the Pfaffian is either of additive Hankel composition or of truncated Wiener-Hopf type. Akhiezer-Kac asymptotic results for the Fredholm Pfaffians are then derived as natural consequences of the Riemann-Hilbert characterisation.