arXiv 2103.119472021-03-22math.PRmath.CVmath.FA

高斯幂级数的零点、Hardy空间和行列式点过程

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Safari Mukeru, Mmboniseni P. Mulaudzi

摘要：

给定一个标准同分布复高斯随机变量序列$(ξ_n)$，Peres和Virág（在论文“零的同分布”中），给出了一个新的随机变量序列.高斯幂级数：共形不变行列式过程“{ \it Acta Math.}（2005）194，1-35）发现了一个惊人的事实，即复单位圆$\mathbb{D}$中的随机幂级数$f(z) = \sum_{n=1}^\infty ξ_n z^{n-1}$的零点构成了一个行列式点过程。放松对随机变量$(ξ_n)$独立性的限制后，对一般随机序列$f(z)$零点的研究是一个重要的公开问题。本文证明了：如果$(ξ_n)$是复高斯随机变量的无穷序列，且其协方差阵可逆，其逆矩阵是Toeplitz矩阵，则$f(z)$的零点集构成一个行列式点过程，其分布与Peres和Virág研究的独立同分布变量的情形相同.这些论点是基于Hardy空间和再生核之间的一些相互作用。说明性的例子构造从经典的Toeplitz矩阵和经典的分数高斯噪声。

英文摘要：

Given a sequence $(ξ_n)$ of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper ``Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process'' {\it Acta Math.} (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series $f(z) = \sum_{n=1}^\infty ξ_n z^{n-1}$ in the complex unit disc $\mathbb{D}$ constitute a determinantal point process. The study of the zeros of the general random series $f(z)$ where the restriction of independence is relaxed upon the random variables $(ξ_n)$ is an important open problem. This paper proves that if $(ξ_n)$ is an infinite sequence of complex Gaussian random variables such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of $f(z)$ constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virág. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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