AI 中文总结
研究一种维格纳型随机矩阵，其非对角元素相关性由特定随机因子生成。通过经典展开获积分算子奇异值，随着耦合强度增加，每个奇异值驱动BBP转变，产生离散临界点层次结构，此机制可推广到更广泛相关结构族。
AI 中文摘要
我们研究一种维格纳型随机矩阵，其非对角元素相关性由给定行和列中所有元素共享的随机因子生成，随着矩阵大小增加耦合强度保持固定。尽管主体谱矩仍遵循纯半圆律，但我们表明基础相关矩阵分解为一个消失的主体以及可数个离群特征值族，在固定秩\(k\)时，这些离群特征值收敛于一个紧致沃尔泰拉（累积和）积分算子的奇异值，通过布朗运动的经典卡尔胡宁 - 勒夫展开以封闭形式获得，并通过数值验证前二十个此类值的精度优于百分之一。随着耦合强度增加，每个奇异值驱动一个独立的baik - Ben Arous - Péché（BBP）转变，产生均匀间隔的离散临界点层次结构，而不是单个转变，在每个临界点处有一个进一步的特征值从半圆边缘分离，这与直接对角化密切一致。我们表明这种机制推广到更广泛的相关结构族，在每种情况下临界层次结构由相关紧致积分算子的谱设定。
英文摘要
We study a Wigner-type random matrix in which the off-diagonal correlation between entries is generated by a random factor shared among all entries in a given row and column, with the coupling strength held fixed as the matrix size grows. Although the bulk spectral moments remain those of the pure semicircle law, we show that the underlying correlation matrix decomposes into a vanishing bulk together with a countable family of outlier eigenvalues that, at fixed rank $k$, converge to the singular values of a compact Volterra (cumulative-sum) integral operator -- obtained in closed form via the classical Karhunen--Loève expansion of Brownian motion and confirmed numerically to better than one percent across the top twenty such values. Each singular value drives an independent Baik--Ben Arous--Péché (BBP) transition as the coupling strength increases, producing an evenly spaced, discrete hierarchy of critical points -- rather than a single transition -- at each of which one further eigenvalue detaches from the semicircle edge, in close agreement with direct diagonalization. We show that this mechanism generalizes to a broader family of correlation structures, with the critical hierarchy in every case set by the spectrum of an associated compact integral operator.
Comments26 pages, 2 figures
