### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 018, 43 pages      arXiv:1708.02519      https://doi.org/10.3842/SIGMA.2018.018
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

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

Christophe Charlier a and Alfredo Deaño b
a) Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, SE-114 28 Stockholm, Sweden
b) School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7FS, UK

Received November 02, 2017, in final form February 27, 2018; Published online March 07, 2018

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.

Key words: asymptotic analysis; Riemann-Hilbert problems; Hankel determinants; random matrix theory; Painlevé equations.

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