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 $(i,j)$-entry being the modified Bessel functions of order $i-j-\nu$, $\nu \in \mathbb{C}$. When the degree $n$ is finite, we show that the Toeplitz determinant is described by the isomonodromy $\tau$-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 $\nu +\frac{1}{2}$. 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 $\psi$-function of the Jimbo-Miwa Lax pair for the inhomogeneous Painlevé II equation is involved.

arXiv comment41 pages, 14 figures

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Cited by in corpus (1)

• Asymptotics of the partition function of the perturbed Gross-Witten-Wadia unitary matrix model