图核矩阵特征值的分布极限
Distributional Limits for Eigenvalues of Graphon Kernel Matrices
AI总结：
本文研究了图核矩阵特征值的分布极限，揭示了非退化和退化情形下的不同波动行为，扩展了经典算子级论证的范围。
摘要：
我们研究了由密集图核随机图产生的核矩阵中个体特征值的波动行为。在图核的最小积分性和有界性假设下，我们建立了相关积分算子简单且分离特征值的分布极限。出现了一个明确的概率二元性：在非退化情形下，适当归一化的经验特征值满足具有显式方差的中心极限定理；而在退化情形下，主导的随机项消失，中心化特征值收敛于由算子谱决定的加权卡方分布。分析不需要对核施加光滑性或利普希茨条件。先前在类似假设下仅建立了算子收敛和特征空间一致性；当前结果刻画了个体特征值的完整分布行为，将波动理论扩展到了经典算子级论证的范围之外。证明结合了二阶扰动展开、核矩阵的集中界以及对称统计的霍夫丁分解，揭示在√n尺度下，主导的随机性来源于潜在位置采样而非伯努利边噪声。
英文摘要：
We study eigenvalue fluctuations of kernel and adjacency matrices in dense graphon-based random graph models, for modes associated with simple, isolated nonzero eigenvalues of the graphon integral operator. Our results require no smoothness or Lipschitz continuity. They include degenerate non-leading modes whose normalized eigenfunctions have constant modulus and change sign between $-1$ and $+1$ on measurable sets. Continuous graphons exclude this possibility: their nonzero eigenfunctions have continuous representatives, so constant modulus forces a constant eigenfunction, which corresponds to the largest eigenvalue of a nonnegative graphon. Nevertheless, non-leading degeneracy already occurs at the second eigenvalue of a two-block model with equal blocks. The limiting behavior is governed by the variance of the squared population eigenfunction. When this variance is positive, the empirical eigenvalue, divided by $n-1$ and centered at its population value, has a Gaussian limit under $\sqrt{n}$ scaling. When the variance vanishes, the first-order fluctuation disappears, and the centered unnormalized kernel eigenvalue converges to a weighted series of centered chi-square variables. For adjacency matrices, Bernoulli edge noise is negligible at the $\sqrt{n}$ scale of the unnormalized eigenvalues. In the degenerate regime, after the same deterministic centering used for the kernel eigenvalue, the adjacency eigenvalue converges to the same weighted chi-square series plus a deterministic shift and a Gaussian term independent of that series. The proofs combine spectral localization, exact Schur identities, finite-mode approximations, and Hoeffding decompositions.
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