Hardy--Szegő 零点窗口计数的泛函中心极限定理
A Functional Central Limit Theorem for Window Counts of Hardy--Szegő Zeros
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中文总结 AI 辅助
本文研究上半平面中 Hardy--Szegő 零点过程，证明扩张水平窗口内中心化零点计数的泛函中心极限定理，并推导协方差密度、渐近方差及白噪声极限。
中文摘要 AI 辅助
Hardy--Szegő 零点过程是 Peres 和 Virág 通过独立同分布高斯解析函数在圆盘上研究的一个典型共形不变行列式点过程。本文研究其上半平面实现。尽管与圆盘模型共形等价，该实现具有其自身的自然几何：实平移不变性使过程沿边界成为平稳对象，并使长水平窗口成为自然可观测量。对于每个容许的高度窗口，我们证明了在扩张的水平窗口中中心化零点计数的 Donsker 型泛函中心极限定理，其显式强度和方差依赖于高度窗口。证明基于阶乘累积量和 Brillinger 混合。主要技术输入是对约化累积量密度的一族全阶可积性估计，通过在高度变量积分之前利用行列式循环结构获得。作为进一步推论，我们推导出显式协方差密度、渐近方差公式以及线性统计量的宏观高斯白噪声极限。
英文摘要
The Hardy--Szegő zero process, investigated in the disk by Peres and Virág through the independent identically distributed Gaussian analytic function, is a canonical conformally invariant determinantal point process. This paper studies its upper half-plane realization. Although conformally equivalent to the disk model, this realization has its own natural geometry: real-translation invariance turns the process into a stationary object along the boundary and makes long horizontal windows the natural observables. For every admissible height window, we prove a Donsker-type functional central limit theorem for the centered zero counts in expanding horizontal windows, with an explicit intensity and variance depending on the height window. The proof is based on factorial cumulants and Brillinger mixing. The main technical input is a family of all-order integrability estimates for reduced cumulant densities, obtained by exploiting the determinantal cycle structure before integrating over the height variables. As further consequences, we derive an explicit covariance density, an asymptotic variance formula, and a macroscopic Gaussian white-noise limit for linear statistics.
发表机构
- Soochow University（苏州大学）
- Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
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