IMRN-2019-925.R1_Proof_hi.pdf (880.55 kB)
Characteristic polynomials of complex random matrices and Painlevé transcendents
journal contribution
posted on 2023-06-09, 20:45 authored by Alfredo Deaño, Nicholas SimmNicholas SimmWe 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?8. 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.
Funding
Random matrix theory and log-correlated Gaussian fields; G2472; ROYAL SOCIETY; URF\R1\180707
History
Publication status
- Published
File Version
- Accepted version
Journal
International Mathematics Research NoticesISSN
1073-7928Publisher
Oxford University PressExternal DOI
Department affiliated with
- Mathematics Publications
Research groups affiliated with
- Probability and Statistics Research Group Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2020-03-03First Open Access (FOA) Date
2021-05-26First Compliant Deposit (FCD) Date
2020-03-01Usage metrics
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