Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge
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 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.
Keywords
Cite
@article{arxiv.1507.01710,
title = {Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge},
author = {Alexander Bogatskiy and Tom Claeys and Alexander Its},
journal= {arXiv preprint arXiv:1507.01710},
year = {2016}
}
Comments
35 pages, 4 figures