Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Fredholm Pfaffians and Riemann-Hilbert problems

Thomas Bothner, Amari Jaconelli

TL;DR

This work develops an analytic framework to compute Fredholm Pfaffians via commutation and Riemann–Hilbert methods for kernels with either additive Hankel composition or truncated Wiener–Hopf structure. By splitting the main kernel into symplectic-derived or orthogonal-derived forms and exploiting commutation identities, the authors reduce complex Pfaffians to combinations of Fredholm determinants of simpler operators, which can be characterized by canonical RHPs. They derive explicit Hankel- and Wiener–Hopf-type formulas (c3, c7, c11, c16, c19, c20, c23, c26, c33, c34) and establish nonlinear steepest-descent analyses (via HankelRHP and WHRHP) to obtain Akhiezer–Kac-type asymptotics as the parameter t grows, connecting to classical KT-K and FTZ-type results. The paper also discusses multiplicative Hankel-type kernels and clarifies the limitations and special structure required for such kernels to fit the framework. Overall, the results furnish a coherent, RHP-grounded path from Fredholm Pfaffians to explicit asymptotics for broad kernel classes relevant in random matrix theory and integrable systems.

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.

On Fredholm Pfaffians and Riemann-Hilbert problems

TL;DR

This work develops an analytic framework to compute Fredholm Pfaffians via commutation and Riemann–Hilbert methods for kernels with either additive Hankel composition or truncated Wiener–Hopf structure. By splitting the main kernel into symplectic-derived or orthogonal-derived forms and exploiting commutation identities, the authors reduce complex Pfaffians to combinations of Fredholm determinants of simpler operators, which can be characterized by canonical RHPs. They derive explicit Hankel- and Wiener–Hopf-type formulas (c3, c7, c11, c16, c19, c20, c23, c26, c33, c34) and establish nonlinear steepest-descent analyses (via HankelRHP and WHRHP) to obtain Akhiezer–Kac-type asymptotics as the parameter t grows, connecting to classical KT-K and FTZ-type results. The paper also discusses multiplicative Hankel-type kernels and clarifies the limitations and special structure required for such kernels to fit the framework. Overall, the results furnish a coherent, RHP-grounded path from Fredholm Pfaffians to explicit asymptotics for broad kernel classes relevant in random matrix theory and integrable systems.

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.

Paper Structure

This paper contains 13 sections, 26 theorems, 282 equations.

Table of Contents

  1. Introduction
  2. Main objectives and first results
  3. Hankel composition type and second results
  4. Truncated Wiener-Hopf type and third results
  5. Proof of \ref{['c3']} and \ref{['c7']} via commutation
  6. Hankel composition type kernels - proof of \ref{['c11']},\ref{['c16']} and \ref{['c19']},\ref{['c20']}
  7. Asymptotic resolution of RHP \ref{['HankelRHP']}
  8. Derivation of asymptotics for $D_M(t)$ and $D_{M,2}(t)$
  9. Wiener-Hopf type kernels - proof of \ref{['c23']},\ref{['c26']} and \ref{['c30']},\ref{['c31']}, as well as \ref{['c33']},\ref{['c34']}
  10. Asymptotic resolution of RHP \ref{['WHRHP']}
  11. Derivation of asymptotics for $D_M(t)$ and $D_{M,2}(t)$
  12. Comment about multiplicative Hankel composition kernels
  13. One useful definition and one useful tool

Key Result

Proposition 2.2

Let $D_M(\Delta)$ denote the Fredholm determinant of $M$ in z5 on $L^2(\Delta)\oplus L^2(\Delta)$, with the aforementioned assumptions placed on $M_{ij}$. Then we have the identity provided $I-2S$ is invertible on $L^2(\Delta)$ and where $D_{2S}(\Delta)$ is the Fredholm determinant of the trace class operator $2S$ acting on $L^2(\Delta)$.

Theorems & Definitions (65)

  • Remark 1.1
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Corollary 3.1
  • Corollary 3.2
  • Theorem 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 55 more