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Bessel kernel determinants and integrable equations

Giulio Ruzza

TL;DR

This work analyzes multiplicative statistics of the Bessel determinantal point process through a Riemann–Hilbert/IIKS integrable-operator framework. It derives a nonlinear integrable PDE governing $v(x,t)$, linked to isospectral deformations of a Sturm–Liouville operator, and establishes an integro-differential Painlevé V structure that encompasses both the Tracy–Widom and Charlier–Doeraene PV reductions. Special Bessel-based solutions are constructed via Darboux transformations, with probabilistic interpretations in terms of Jánossy densities for thinned Bessel processes. The results provide a comprehensive connection between Fredholm determinant statistics, integrable systems, and Painlevé transcendents, with a boundary-value problem formulation enabling explicit initial-data mappings.

Abstract

We derive differential equations for multiplicative statistics of the Bessel determinantal point process depending on two parameters. In particular, we prove that such statistics are solutions to an integrable nonlinear partial differential equation describing isospectral deformations of a Sturm-Liouville equation. We also derive identities relating solutions to the integrable partial differential equation and to the Sturm-Liouville equation which imply an analogue for Painlevé V of Amir-Corwin-Quastel "integro-differential Painlevé II equation". This equation reduces, in a degenerate limit, to the system of coupled Painlevé V equations derived by Charlier and Doeraene for the generating function of the Bessel process, and to the Painlevé V equation derived by Tracy and Widom for the gap probability of the Bessel process. Finally, we study an initial value problem for the integrable partial differential equation. The approach is based on Its-Izergin-Korepin-Slavnov theory of integrable operators and their associated Riemann-Hilbert problems.

Bessel kernel determinants and integrable equations

TL;DR

This work analyzes multiplicative statistics of the Bessel determinantal point process through a Riemann–Hilbert/IIKS integrable-operator framework. It derives a nonlinear integrable PDE governing , linked to isospectral deformations of a Sturm–Liouville operator, and establishes an integro-differential Painlevé V structure that encompasses both the Tracy–Widom and Charlier–Doeraene PV reductions. Special Bessel-based solutions are constructed via Darboux transformations, with probabilistic interpretations in terms of Jánossy densities for thinned Bessel processes. The results provide a comprehensive connection between Fredholm determinant statistics, integrable systems, and Painlevé transcendents, with a boundary-value problem formulation enabling explicit initial-data mappings.

Abstract

We derive differential equations for multiplicative statistics of the Bessel determinantal point process depending on two parameters. In particular, we prove that such statistics are solutions to an integrable nonlinear partial differential equation describing isospectral deformations of a Sturm-Liouville equation. We also derive identities relating solutions to the integrable partial differential equation and to the Sturm-Liouville equation which imply an analogue for Painlevé V of Amir-Corwin-Quastel "integro-differential Painlevé II equation". This equation reduces, in a degenerate limit, to the system of coupled Painlevé V equations derived by Charlier and Doeraene for the generating function of the Bessel process, and to the Painlevé V equation derived by Tracy and Widom for the gap probability of the Bessel process. Finally, we study an initial value problem for the integrable partial differential equation. The approach is based on Its-Izergin-Korepin-Slavnov theory of integrable operators and their associated Riemann-Hilbert problems.
Paper Structure (21 sections, 19 theorems, 155 equations)

This paper contains 21 sections, 19 theorems, 155 equations.

Table of Contents

  1. Introduction and results
  2. Remarks on Theorem \ref{['thm']}
  3. Uniqueness of f for alpha positive
  4. Integrability
  5. Special solutions
  6. Relation to Painlevé V and to a system of coupled Painlevé V equations
  7. A boundary value problem
  8. Probability and operator preliminaries
  9. The Bessel kernel operator
  10. Multiplicative statistics of the Bessel process
  11. Full gap probability of a thinned Bessel process
  12. Positive-temperature deformation of the Bessel process
  13. Riemann--Hilbert problems
  14. Integrable operators
  15. Bessel dressing
  16. ...and 6 more sections

Key Result

Theorem 1.2

Under the hypotheses in Assumption assumption, the following facts hold true.

Theorems & Definitions (49)

  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 39 more