  In this paper, we consider polynomials orthogonal with respect to a varying
perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain
contours in the complex plane. When the parameters $n$, $t$ and the degree $k$
are fixed, the Hankel determinant for the singular complex weight is shown to
be the isomonodromy $\tau$-function of the Painlev\'e III equation. When the
degree $k=n$, $n$ is large and $t$ 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 tronqu\'ee solution to
the Painlev\'e I equation. Our approach is based on the Deift-Zhou nonlinear
steepest descent method for Riemann-Hilbert problems.
