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Characteristic polynomials of complex random matrices and Painlevé transcendents

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Análisis Aplicadoes
dc.contributor.authorDeaño Cabrera, Alfredo
dc.contributor.authorSimm, Nick
dc.date.accessioned2022-01-10T12:37:04Z
dc.date.available2022-12-01T00:00:05Z
dc.date.issued2021-12
dc.description.abstractWe study expectations of powers and correlation functions for characteristic polynomials of N×N non-Hermitian random matrices. For the 1-point and 2-point correlation function, we obtain several characterizations in terms of Painlevé transcendents, both at finite N and asymptotically as N→∞⁠. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginibre ensemble this involves Painlevé IV at the boundary as N→∞⁠. Our approach, together with the results in [ 49], suggests that this should arise in a much broader class of planar models. For the bulk asymptotics, one of our results can be interpreted as the merging of two “planar Fisher–Hartwig singularities” where Painlevé V arises in the asymptotics. We also discuss the correspondence of our results with a normal matrix model with d-fold rotational symmetries known as the lemniscate ensemble, recently studied in [ 15, 18]. Our approach is flexible enough to apply to non-Gaussian models such as the truncated unitary ensemble or induced Ginibre ensemble; we show that in the former case Painlevé VI arises at finite N⁠. Scaling near the boundary leads to Painlevé V, in contrast to the Ginibre ensemble.en
dc.format.extent55
dc.identifier.bibliographicCitationDeaño, A. & Simm, N. (2020). Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents. International Mathematics Research Notices, 2021(24), 19000–19054.en
dc.identifier.doihttps://doi.org/10.1093/imrn/rnaa111
dc.identifier.issn1073-7928
dc.identifier.publicationfirstpage19000
dc.identifier.publicationissue24
dc.identifier.publicationlastpage19054
dc.identifier.publicationtitleInternational Mathematics Research Noticesen
dc.identifier.publicationvolume2021
dc.identifier.urihttps://hdl.handle.net/10016/33852
dc.identifier.uxxiAR/0000026488
dc.language.isoengen
dc.publisherOxford University Pressen
dc.rights© The Author(s) 2018. Published by Oxford University Press. All rights reserved.en
dc.rights.accessRightsopen accessen
dc.subject.ecienciaMatemáticases
dc.titleCharacteristic polynomials of complex random matrices and Painlevé transcendentsen
dc.typeresearch article*
dc.type.hasVersionAM*
dspace.entity.typePublication
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