Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

Abstract

We study $n\times n$ Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large $n$ asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with $n$. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlevé IV equation.

References in corpus (3)

• On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials
• The birth of a cut in unitary random matrix ensembles
• Large-degree asymptotics of rational Painleve-IV functions associated to generalized Hermite polynomials

Cited by in corpus (6)

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• Full counting statistics for interacting trapped fermions
• Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants
• Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
• Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics
• Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics