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

TL;DR

This paper analyzes the asymptotic behavior of Hankel determinants and orthogonal polynomials with a Gaussian weight featuring a near-edge discontinuity, revealing connections to Painlevé II solutions and applications in random matrix theory.

Contribution

It provides new asymptotic formulas for Hankel determinants with discontinuous Gaussian weights near the edge, linking them to Painlevé II solutions and exploring their implications.

Findings

Asymptotics described by Ablowitz-Segur solutions to Painlevé II

Conjectures on Airy kernel Fredholm determinants

New results on poles of Painlevé II solutions

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.