标题： 修正贝塞尔函数行列式的渐近性和第二 Painlevé 方程 显示英文标题

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

摘要： 在本文中，我们通过在势能中引入一个对数项来考虑扩展的Gross-Witten-Wadia酉矩阵模型。 该模型的分区函数可以等价地用托普利茨行列式表示，其中$(i,j)$项是阶数为$i-j-\nu$、$\nu\in\mathbb{C}$的修正贝塞尔函数。 当次数$n$为有限时，我们证明托普利茨行列式由Painlevé III方程的单形变$\tau$函数描述。 作为双重标度极限，%在次数$n\to\infty$的双重标度极限下，我们建立了托普利茨行列式对数导数的渐近逼近，该逼近用带有参数$\nu+\frac{1}{2}$的非齐次Painlevé II方程的Hastings-McLeod解表示。 相关正交多项式的主系数和递推系数的渐近行为也得到了推导。 我们通过将Deift-Zhou非线性最陡下降方法应用于Hankel环上的正交多项式的黎曼-希尔伯特问题来获得这些结果。 这里的主要问题是临界点$z=-1$处局部参数化的构造，其中涉及非齐次 Painlevé II 方程的 Jimbo-Miwa Lax 对的$\psi$函数。 显示英文摘要

摘要： 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 Painlev\'{e} 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 Painlev\'{e} 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 Painlev\'{e} II equation is involved.