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/μ-∑j=1∞cj/zj, where each cj 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↦ 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 | μ| → ∞ is a determinantal point process obtaine..