Hankel determinants for a singular complex weight and the first and third Painlevé transcendents

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight e−n(z−logz+t/z)e−n(z−logz+t/z) for t<0t<0 and zz on certain contours in the complex plane. When the parameters nn, tt and the degree kk are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy ττ-function of the Painlevé III equation. When the degree k=nk=n, nn is large and tt is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients and the Hankel determinant are described in terms of a Boutroux tronquée solution to the Painlevé I equation. Our approach is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.