Characteristic polynomials of complex random matrices and Painlevé transcendents

Deaño, Alfredo and Simm, Nick (2020) Characteristic polynomials of complex random matrices and Painlevé transcendents. International Mathematics Research Notices. ISSN 1073-7928

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Abstract

We 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 \to \infty. Our approach, together with the results in \cite{HW17} 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 \textit{lemniscate ensemble}, recently studied in \cite{BGM, BGG18}. 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.

Item Type: Article
Keywords: random matrix, Ginibre ensemble, Coulomb gas, planar models
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Probability and Statistics Research Group
SWORD Depositor: Mx Elements Account
Depositing User: Mx Elements Account
Date Deposited: 03 Mar 2020 11:23
Last Modified: 30 Mar 2022 16:48
URI: http://sro.sussex.ac.uk/id/eprint/90209

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Project NameSussex Project NumberFunderFunder Ref
Random matrix theory and log-correlated Gaussian fieldsG2472ROYAL SOCIETYURF\R1\180707