Extreme principal minors of Wishart and deformed GOE matrices

Abstract

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.