AI 中文总结
该研究针对逐元素变换的加尖Wigner矩阵，证明无导数的线性谱统计定理，明确其均值、协方差结构及相关准则，揭示单纯L^4(ν)不满足要求。
AI 中文摘要
我们针对具有一般微观噪声律、经逐元素变换的秩一加尖Wigner矩阵，证明了一个无导数的解析线性谱统计定理。该变换需满足中心化、方差归一化，且在加尖产生的微小平移下具有容许性：其平移后的均值与二阶矩对应一阶和二阶剖面，而其中心化平移后的四阶累积量与四阶尾项具有稳定性。在每一个微观平移处，我们构造了一个显式的一致有界三点变量，其恰好匹配目标逐元素的前四个中心化矩。广义Wigner线性谱统计定理适用于所得的有界三角阵列，且全局傅里叶-杜哈梅尔导数估计将解析统计量传递回粗糙变换，无需共同截断、系数稳定性假设或局部律。一阶均值包含齐次Wigner偏差、秩一Woodbury响应、零对角校正项以及二次方差剖面响应。中心化协方差为标准零对角实Wigner协方差，其四阶累积量参数为κ₄^{f,ν}。我们将体轮廓统计量与超临界区的全迹区分开，证明分离的异常值恰好向全迹中心化项添加φ(θ+θ⁻¹)。作为自包含推论，当f∈L^{4+ε}(γ)时，高斯噪声无需f可微即可满足该定理；所得的体与全迹推论具有显式Hermite系数，包含所有中心化、方差归一化的多项式增长变换。我们提供了具体的似然比、平滑变换、有界粗糙变换及原子准则，同时给出阻碍条件，表明单纯的L^4(ν)是不够的。
英文摘要
We prove a derivative-free analytic linear spectral statistics theorem for entrywise-transformed rank-one spiked Wigner matrices with a general microscopic noise law. The transform is required to be centered, variance-normalized, and admissible under the small translations generated by the spike: its shifted mean and second moment have first- and second-order profiles, while its centered shifted fourth cumulants and fourth tails are stable. At every microscopic shift we construct an explicit uniformly bounded three-point variable matching the first four centered moments of the target entry exactly. A generalized-Wigner LSS theorem applies to the resulting bounded triangular array, and a global Fourier--Duhamel derivative estimate transfers the analytic statistic back to the rough transform without a common truncation, coefficient-stability assumption, or local law. The order-one mean consists of the homogeneous Wigner bias, a rank-one Woodbury response, a zero-diagonal correction, and a quadratic variance-profile response. The centered covariance is the standard zero-diagonal real-Wigner covariance with fourth-cumulant parameter $κ_4^{f,ν}$. We distinguish the bulk contour statistic from the full trace in the supercritical regime and show that the separated outlier adds exactly $φ(θ+θ^{-1})$ to the full-trace centering. As a self-contained consequence, Gaussian noise with $f\in L^{4+ε}(γ)$ satisfies the theorem without differentiability of $f$; the resulting bulk and full-trace corollary has explicit Hermite coefficients and includes every centered, variance-normalized polynomial-growth transform. Concrete likelihood-ratio, smooth-transform, bounded rough-transform, and atomic criteria are provided, together with obstructions showing that bare $L^4(ν)$ is insufficient.
Comments56 pages; Comments are welcome
