Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities

Charlier, Christophe
(2019) International Mathematics Research Notices — Vol. rny009 (2018)

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We obtain asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher–Hartwig singularities. This generalises two results: (1) a result of Berestycki, Webb, and Wong [5] for root-type singularities and (2) a result of Its and Krasovsky [37] for a Gaussian weight with a single jump-type singularity. We show that when we apply a piecewise constant thinning on the eigenvalues of a random Hermitian matrix drawn from a one-cut regular ensemble, the gap probability in the thinned spectrum, as well as correlations of the characteristic polynomial of the associated conditional point process, can be expressed in terms of these determinants.
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Charlier, C. (2019). Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities. International Mathematics Research Notices, rny009. https://doi.org/10.1093/imrn/rny009 (Original work published 2018)