Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge
arXiv:1507.01710 · doi:10.1007/s00220-016-2691-y
Abstract
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 Painlevé 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 Painlevé II transcendents, and we also prove a new result on the poles of the Ablowitz-Segur solutions to the Painlevé II equation. We also highlight applications of our results in random matrix theory.
35 pages, 4 figures