  We compute asymptotics for Hankel determinants and orthogonal polynomials
with respect to a discontinuous Gaussian weight, in a critical regime where the
discontinuity is close to the edge of the associated equilibrium measure
support. Their behavior is described in terms of the Ablowitz-Segur family of
solutions to the Painlev\'e II equation. Our results complement the ones in
[Xu,Zhao,2011]. As consequences of our results, we conjecture asymptotics for
an Airy kernel Fredholm determinant and total integral identities for
Painlev\'e II transcendents, and we also prove a new result on the poles of the
Ablowitz-Segur solutions to the Painlev\'e II equation. We also highlight
applications of our results in random matrix theory.
