高斯Weyl-Heisenberg函数零点的超均匀性与非超均匀性
Hyperuniformity and non-hyperuniformity of zeros of Gaussian Weyl-Heisenberg Functions
AI总结：
研究复平面上扭曲平稳高斯随机函数的零点统计，发现带电荷零点方差线性增长（超均匀），而不带电荷零点方差可能随面积增长（非超均匀），并应用于谱图零点统计优于极值点的理论证明。
摘要：
我们研究了复平面上扭曲平稳高斯随机函数的零点集，即在Weyl-Heisenberg群作用下随机平稳的高斯随机函数。该模型包括平移不变的高斯整函数（GEFs），以及许多其他非解析的例子，在这些例子中，零点周围的缠绕数可以是正的或负的。我们研究了零点按其缠绕数加权（带电零点集）和未加权（不带电零点集）时的零点统计。我们证明了带电零点统计的方差总是随观测圆盘半径线性增长（超均匀性）。重要的是，这对于可能具有非零均值的函数成立，且无需假设额外的对称性（如径向性）。对于不带电零点统计，我们提供了一个例子，其中方差随观测圆盘面积增长（非超均匀性）。这用于证明，虽然GEFs的零点是超均匀的，但其临界点集却不是。我们的工作对统计信号处理的最新发展有所贡献，其中嵌入噪声的非平稳信号的时频分布通过对其谱图零点（“静默点”）进行统计检验来揭示。我们证明了经验谱图零点计数在大观测窗口内与其系综平均具有适度偏差（这一点以前仅对纯噪声已知）。相反，我们还证明了谱图极大值（“响亮点”）不具备类似性质。这首次为静默点相对于竞争特征响亮点的统计优越性提供了形式证据，这一事实已被实践者注意到。
英文摘要：
We study zero sets of twisted stationary Gaussian random functions on the complex plane, i.e., Gaussian random functions that are stochastically invariant under the action of the Weyl-Heisenberg group. This model includes translation-invariant Gaussian entire functions (GEFs), and also many other non-analytic examples, in which case winding numbers around zeros can be either positive or negative. We investigate zero statistics both when zeros are weighted with their winding numbers (charged zero set) and when they are not (uncharged zero set). We show that the variance of the charged zero statistic always grows linearly with the radius of the observation disk (hyperuniformity). Importantly, this holds for functions with possibly non-zero means and without assuming additional symmetries such as radiality. With respect to uncharged zero statistics, we provide an example for which the variance grows with the area of the observation disk (non-hyperuniformity). This is used to show that, while the zeros of GEFs are hyperuniform, the set of their critical points fails to be so. Our work contributes to recent developments in statistical signal processing, where the time-frequency profile of a non-stationary signal embedded into noise is revealed by performing a statistical test on the zeros of its spectrogram ("silent points"). We show that empirical spectrogram zero counts enjoy moderate deviations from their ensemble averages over large observation windows (something that was previously known only for pure noise). In contrast, we also show that spectrogram maxima ("loud points") fail to enjoy a similar property. This gives the first formal evidence for the statistical superiority of silent points over the competing feature of loud points, a fact that has been noted by practitioners.
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