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

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 is an element of C. When the degree n is finite, we show that the Toeplitz determinant is described by the isomonodromy tau-function of the Painleve 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 Painleve II equation with parameter nu + 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 Painleve II equation is involved.