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

• 成果类型: Article
• 署名作者: Forrester, Peter J.; Ipsen, Jesper R.
• 署名单位: University of Melbourne
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00903-7
• 发表日期: 2019
• 页码: 833-847
• 关键词: exact statistics

摘要

The zeros of the random Laurent series 1/mu - 8 j=1 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 . 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|. 8 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.