Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge

Alexander Bogatskiy, Alexander Its, Tom Claeys

Published 2015-07-07Version 1

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 from 2011 by Xu and Zhao. As consequences of our results, we obtain asymptotics for an Airy kernel Fredholm determinant, total integral identities for Painlev\'e II transcendents, and 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.