Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump

TL;DR

This paper derives asymptotic formulas for a Hankel determinant with a Gaussian weight modified by a step function, using Riemann-Hilbert analysis of orthogonal polynomials.

Contribution

It introduces a novel asymptotic analysis of Hankel determinants with jump discontinuities in the Gaussian weight via Riemann-Hilbert methods.

Findings

Asymptotic behavior of the Hankel determinant for large n

Explicit formulas for orthogonal polynomials with jump weights

Application of Riemann-Hilbert analysis to jump-modified weights

Abstract

We obtain asymptotics in n for the n-dimensional Hankel determinant whose symbol is the Gaussian multiplied by a step-like function. We use Riemann-Hilbert analysis of the related system of orthogonal polynomials to obtain our results.