Wishart矩阵与变形GOE矩阵的极端主子式
Extreme principal minors of Wishart and deformed GOE matrices
浏览论文内容
中文总结 AI 辅助
该研究针对Wishart与变形GOE矩阵的极端主子式问题，提出基于Hausdorff距离收敛的确定性集合识别方法，解决了二阶矩论证未覆盖区域的问题，确定了变形GOE矩阵的相变及Wishart矩阵的极限。
中文摘要 AI 辅助
我们研究Wishart矩阵与变形GOE矩阵所有主子式中最大特征值的大数定律。我们提出一种新方法，该方法基于确定确定性集合——经适当归一化的主子式构成的随机集合在Hausdorff距离下收敛到这些确定性集合，从而将原极端值问题简化为有限维凸优化问题。我们证明该方法在现有二阶矩论证（见文献[cai2021asymptotic, hu2023extreme]）未覆盖的区域有效。对于固定子式大小为k的变形GOE矩阵，我们确定了所有对角方差a>0的极限，并识别出a=2处的相变：相变之上，极限常数满足无闭式表达式的显式递归，且优化器呈现嵌套层次结构，从而解决了文献[cai2021asymptotic]中遗留的情况。对于具有一般次高斯项且固定k的Wishart矩阵，我们通过熵约束的确定性凸集刻画其极限；当项为标准高斯时，我们显式求解所得优化问题，得到极限常数的精确值。
英文摘要
We study the laws of large numbers for the largest eigenvalues among all principal minors of Wishart matrices and deformed GOE matrices. We propose a new method based on identifying the deterministic sets to which the random sets formed by suitably normalized principal minors converge in Hausdorff distance, thereby reducing the original extreme-value problems to finite-dimensional convex optimization problems. We demonstrate the effectiveness of this method in regimes not covered by the existing second-moment arguments in \cite{cai2021asymptotic,hu2023extreme}. For deformed GOE matrices with fixed minor size \(k\), we determine the limit for every diagonal variance \(a>0\) and identify a phase transition at \(a=2\). Above the transition, the limiting constant satisfies an explicit recursion with no close-form expression, and the optimizers exhibit a nested hierarchical structure, thereby resolving the case left open in \cite{cai2021asymptotic}. For Wishart matrices with general sub-Gaussian entries and fixed \(k\), we characterize the limit through an entropy-constrained deterministic convex set. When the entries are standard Gaussian, we solve the resulting optimization problem explicitly and obtain the exact value of the limiting constant.
