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 $i-j-\nu$, $\nu \in ℂ$. 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, 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.

在本文中，我们通过在势中引入对数项来考虑扩展的 Gross-Witten-Wadia 酉矩阵模型。模型的配分函数可以用 Toeplitz 行列式等效地表示为$（我,j）$-条目是阶的修正贝塞尔函数$我-j-\nu$,$\nu \epsilon ℂ$。当学位$n$是有限的，我们证明托普利茨行列式由等单律描述$\tau$-Painlevé III 方程的函数。作为双标度极限，我们建立了 Toeplitz 行列式的对数导数的渐近近似，用参数为非齐次 Painlevé II 方程的 Hastings-McLeod 解来表示$\nu +\frac{1}{2}$。还推导了相关正交多项式的首项系数和递推系数的渐近性。我们通过将 Deift-Zhou 非线性最速下降法应用于汉克尔环上正交多项式的黎曼-希尔伯特问题来获得结果。这里主要关注的是关键点处局部参数矩阵的构造$z=-1$，其中$\psi$涉及非齐次 Painlevé II 方程的 Jimbo-Miwa Lax 对的函数。