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

Abstract

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.