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

Christophe Charlier ; Alfredo Deaño

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

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.

Détail

Keywords: asymptotic analysis; Riemann–Hilbert problems; Hankel determinants; random matrix theory; Painlevé equations.

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