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

We study n×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.