Probability Theory and Related Fields

, Volume 175, Issue 3–4, pp 833–847 | Cite as

A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices

  • Peter J. ForresterEmail author
  • Jesper R. Ipsen


The zeros of the random Laurent series \(1/\mu - \sum _{j=1}^\infty c_j/z^j\), where each \(c_j\) is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex Gaussian matrix. For the corresponding random Maclaurin series obtained by the replacement \(z \mapsto 1/z\), we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix. Since the correlation functions of the latter are known, by taking an appropriate limit the correlation functions for the random Maclaurin series can be determined. Only for \(|\mu | \rightarrow \infty \) is a determinantal point process obtained. For the one and two point correlations, by regarding the Maclaurin series as the limit of a random polynomial, a direct calculation can also be given.

Mathematics Subject Classification

26C10 15B52 30B20 



The work is part of a research program supported by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers. PJF also acknowledges partial support from the Australian Research Council Grant DP170102028.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statistical FrontiersThe University of MelbourneMelbourneAustralia

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