Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909110996303872 |
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| author | Chen, Yu Xu, Shuai-Xia Zhao, Yu-Qiu |
| author_facet | Chen, Yu Xu, Shuai-Xia Zhao, Yu-Qiu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_11233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation Chen, Yu Xu, Shuai-Xia Zhao, Yu-Qiu Mathematical Physics Exactly Solvable and Integrable Systems 33E17, 34M55, 41A60 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. |
| title | Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems 33E17, 34M55, 41A60 |
| url | https://arxiv.org/abs/2402.11233 |