AI 中文总结
研究行独立平稳高斯序列构建的长程相关Wigner型矩阵，明确指数衰减与幂律相关性下的体谱、边缘行为及临界特性，验证相关框架假设并给出数值与理论结果。
AI 中文摘要
我们研究由行独立平稳高斯序列构建的长程相关Wigner型矩阵。对于指数衰减（AR(1)）相关性，体谱密度通过显式组合“ hub 机制”从半圆律发生形变，同时我们验证了矩阵 Dyson 方程框架（MDE）的平坦性与衰减假设，数值证据支持对于指数衰减相关性的任意固定 ρ<1，Tracy-Widom 边缘普适性成立；退化极限 ρ→1⁻ 退化为对称 Volterra 算子，与配套 BBP 分析中识别的奇异值级联相联系。对于幂律相关性 dt~t⁻ᵞ，我们确定 γ_c=1/2 是体四阶矩发散的临界点，而 γ=1 标志着支配 MDE 边缘分析的平坦性条件失效。我们严格证明了四阶矩转变，并数值发现自洽边缘在 γ=1 处平滑变化，无任何扭结或不连续性的证据。
英文摘要
We study long-range correlated Wigner-type matrices built from row-independent stationary Gaussian sequences. For exponentially decaying (AR(1)) correlations, the bulk spectral density deforms from the semicircle law via an explicit combinatorial "hub" mechanism, yet we verify the flatness and decay hypotheses of the matrix-Dyson-equation framework (MDE), with numerical evidence supporting Tracy-Widom edge universality for every fixed $ρ<1$ of the exponential decay correlations; the degenerate limit $ρ\to1^-$ reduces to a symmetrized Volterra operator, connecting to the singular-value cascade identified in a companion BBP analysis. For power-law correlations $dt\sim t^{-γ}$, we identify $γ_c=1/2$ as the critical point for divergence of the bulk fourth-moment, while $γ=1$ marks the breakdown of the flatness condition governing the MDE edge analysis. We prove the fourth-moment transition exactly and find numerically that the self-consistent edge varies smoothly across $γ=1$, with no evidence of a kink or discontinuity.
Comments53 pages, 4 figures
