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Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation

Yu Chen, Shuai-Xia Xu, Yu-Qiu Zhao

Abstract

In the paper, we consider the extended Gross-Witten-Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the $(i,j)$-entry being the modified Bessel functions of order $i-j-ν$, $ν\in\mathbb{C}$. When the degree $n$ is finite, we show that the Toeplitz determinant is described by the isomonodromy $τ$-function of the Painlevé III equation. As a double scaling limit, %In the double scaling limit as the degree $n\to\infty$, we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings-McLeod solution of the inhomogeneous Painlevé II equation with parameter $ν+\frac{1}{2}$. The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift-Zhou nonlinear steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point $z=-1$, where the $ψ$-function of the Jimbo-Miwa Lax pair for the inhomogeneous Painlevé II equation is involved.

Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation

Abstract

In the paper, we consider the extended Gross-Witten-Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the -entry being the modified Bessel functions of order , . When the degree is finite, we show that the Toeplitz determinant is described by the isomonodromy -function of the Painlevé III equation. As a double scaling limit, %In the double scaling limit as the degree , we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings-McLeod solution of the inhomogeneous Painlevé II equation with parameter . The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift-Zhou nonlinear steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point , where the -function of the Jimbo-Miwa Lax pair for the inhomogeneous Painlevé II equation is involved.
Paper Structure (27 sections, 6 theorems, 189 equations, 14 figures)

This paper contains 27 sections, 6 theorems, 189 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Riemann-Hilbert problem for the orthogonal polynomials
  4. Riemann-Hilbert problem for orthogonal polynomials and differential identity
  5. Proof of Theorem \ref{['thm1']}
  6. Asymptotics of the Riemann-Hilbert problem for orthogonal polynomials
  7. Normalization: $\widehat{Y}\to T$
  8. Deformation: $T\to S$
  9. Global parametrix: $N$
  10. Local parametrix and the model Riemann-Hilbert problem for $\Psi$
  11. Final transformation $S\to R$ and error estimate
  12. Asymptotics for $\Psi$ as $x\rightarrow+\infty$
  13. Global parametrix on $[-i,i]$
  14. Local parametrices near $\pm i$
  15. Final transformation
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\nu\in\mathbb{C}$ and $\pi_{n}$ be the $n$-th monic orthogonal polynomials defined by eq:Ortho, and Then $a_{n}(t)$ satisfies the Painlevé III equation Moreover, we have where $c$ is a constant and $\tau(t)$ is the Jimbo-Miwa-Ueno isomonodromy $\tau$-function of the Painlevé III equation PIIIequ.

Figures (14)

  • Figure 1: The Hankel loop $\Gamma$
  • Figure 2: Contours for the transformation $Y\to \widehat{Y}$
  • Figure 3: The contours $\Sigma_{\widehat{Y}}$ of RH problem for $\widehat{Y}$
  • Figure 4: Contours and regions for the RH problem for $S(z)$
  • Figure 5: Contours and regions for the model RH problem
  • ...and 9 more figures

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 2.2
  • Proposition 1
  • proof
  • Remark 3.7
  • ...and 4 more