发表机构
Hangzhou Normal University; Soochow University; National Yang Ming Chiao Tung University; South China University of Technology（杭州师范大学; 苏州大学; 国立阳明交通大学; 华南理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究圆周上临界高斯乘性混沌的高频傅里叶系数渐近，证明其归一化后稳定收敛到条件稳定噪声，并给出几乎必然积分检验，揭示了空间分布而非仅总质量的极限性质。
AI 中文摘要
我们研究了圆周上临界高斯乘性混沌 $M$ 的高频傅里叶系数 $C_n$，针对具有任意可容许光滑、可能非平稳余项的对数协方差。对于每个固定的有限整数偏移集合，相应的相邻系数向量乘以 $\sqrt{\log n}$ 后，相对于完整高斯场稳定收敛，无需额外居中。以该场为条件，极限由指数为一的对称复稳定随机测度的傅里叶系数组成，其控制测度与 $M$ 成比例。对于每个 $s>1$，归一化的调制测度也在 $H^{-s}(\mathbb{T})$ 中稳定收敛到该条件稳定噪声。无条件地，对于每个固定可容许协方差，偏移大小至多 $(\log\log n)^{1/64}$ 时，相同的联合近似仍然有效。标量极限是各向同性柯西混合；联合极限律保留了 $M$ 的空间分布，而不仅仅是其总质量。我们还证明了沿整个整数频率序列的几乎必然积分检验。对于一类正则确定性规范函数 $H$，$\sqrt{\log|n|}\\,|C_n|/H(\log\log|n|)$ 的上极限为零或无穷，取决于 $\int^\infty dx/H(x)$ 收敛或发散。证明结合了指数为一的补偿与相对游程估计，并通过光滑协方差比较将这些结论从典范模型转移过来。
英文摘要
We study the high-frequency Fourier coefficients $C_n$ of critical Gaussian multiplicative chaos $M$ on the circle, for logarithmic covariances with an arbitrary admissible smooth, possibly nonstationary remainder. For every fixed finite set of integer offsets, the corresponding vector of neighboring coefficients, multiplied by $\sqrt{\log n}$, converges stably relative to the full Gaussian field without additional centering. Conditionally on that field, the limit consists of Fourier coefficients of one symmetric complex stable random measure of index one with control proportional to $M$. For every $s>1$, the normalized modulated measures also converge stably in $H^{-s}(\mathbb{T})$ to this conditional stable noise. Unconditionally, the same joint approximation remains valid for offsets of size at most $(\log\log n)^{1/64}$ for every fixed admissible covariance. The scalar limit is an isotropic Cauchy mixture; the joint limiting law retains the spatial distribution of $M$, not only its total mass. We also prove an almost-sure integral test along the entire integer frequency sequence. For a class of regular deterministic gauges $H$, the limsup of $\sqrt{\log|n|}\,|C_n|/H(\log\log|n|)$ is zero or infinity according as $\int^\infty dx/H(x)$ converges or diverges. The proof combines index-one compensation with relative excursion estimates and transfers these conclusions from the canonical model by smooth covariance comparison.
Comments66 pages
