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On the crystal limit of the q-difference sixth Painlevé equation

Nalini Joshi, Pieter Roffelsen

TL;DR

This work analyzes the crystal limit $q\rightarrow 0$ of the Riemann-Hilbert correspondence for the $q$-difference Painlevé VI equation, proving that the limiting map exists and is bi-rational. The authors develop a crystal-limit framework in which the initial value space degenerates to a fixed surface $\mathfrak{X}_t$, the linear problem reduces to a hypergeometric-type system, and the connection data converge to a degree-2 polynomial $C^{\diamond}(z)$. They then describe a crystal-limit Segre surface $\mathcal{F}_t^{\diamond}$ and show that the crystal-limit RH map $\mathrm{RH}_t^{\diamond}$ is an isomorphism between $\mathfrak{X}_t$ and $\mathcal{F}_t^{\diamond}$ with explicit formulas for the image coordinates $\eta^{\diamond}$; the inverse is constructed via Mano-type decompositions. These results reveal a sharp, explicit rational structure emerging in the crystal limit and suggest that similar phenomena may occur for broader classes of Fuchsian $q$-difference systems and their RH problems, with potential extensions to higher-order terms and other scalings.

Abstract

We consider the Riemann-Hilbert correspondence associated with the $q$-difference sixth Painlevé equation in the crystal limit, i.e. $q\rightarrow 0$, and show two main results. First, the limit of this generically highly transcendental mapping is shown to exist. Second, we show that the limiting map is bi-rational and describe it explicitly.

On the crystal limit of the q-difference sixth Painlevé equation

TL;DR

This work analyzes the crystal limit of the Riemann-Hilbert correspondence for the -difference Painlevé VI equation, proving that the limiting map exists and is bi-rational. The authors develop a crystal-limit framework in which the initial value space degenerates to a fixed surface , the linear problem reduces to a hypergeometric-type system, and the connection data converge to a degree-2 polynomial . They then describe a crystal-limit Segre surface and show that the crystal-limit RH map is an isomorphism between and with explicit formulas for the image coordinates ; the inverse is constructed via Mano-type decompositions. These results reveal a sharp, explicit rational structure emerging in the crystal limit and suggest that similar phenomena may occur for broader classes of Fuchsian -difference systems and their RH problems, with potential extensions to higher-order terms and other scalings.

Abstract

We consider the Riemann-Hilbert correspondence associated with the -difference sixth Painlevé equation in the crystal limit, i.e. , and show two main results. First, the limit of this generically highly transcendental mapping is shown to exist. Second, we show that the limiting map is bi-rational and describe it explicitly.
Paper Structure (13 sections, 5 theorems, 141 equations, 1 figure)

This paper contains 13 sections, 5 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Crystal limit of the initial value space and linear problem
  5. Crystal limit of the initial value space
  6. Geometry of the linear system and crystal limit
  7. Crystal limit of canonical solutions
  8. Crystal limit of the connection matrix
  9. A Segre surface and the Riemann-Hilbert correspondence
  10. Tyurin parameters
  11. An affine Segre surface and the Riemann-Hilbert correspondence
  12. Crystal limit of Segre surface and Riemann-Hilbert correspondence
  13. Conclusion

Key Result

Lemma 2.5

The rational mapping realised through the parametrisation $(f,g)\mapsto [A(z)]$, is an isomorphism.

Figures (1)

  • Figure 2.1: Degeneration of base-point configuration in \ref{['eq:basepoints']} under the crystal limit.

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 9 more