Random matrix ensembles with singularities and a hierarchy of Painlevé III equations

Max R. Atkin, Tom Claeys, Francesco Mezzadri

Research output: Contribution to journalArticle (Academic Journal)peer-review

13 Citations (Scopus)
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Abstract

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaus- sian Unitary Ensembles perturbed with a pole of order k at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hier- archy of higher order analogs to the PIII equation, which reduces to the PIII equation itself when the pole is simple.
Original languageEnglish
Pages (from-to)2320-2375
Number of pages56
JournalInternational Mathematics Research Notices
Volume2016
Issue number8
Early online date14 Jul 2015
DOIs
Publication statusPublished - Aug 2016

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