Principal minors of Gaussian orthogonal ensemble

TL;DR

This paper investigates the maximum eigenvalues of principal minors in the Gaussian orthogonal ensemble, showing they follow a Gumbel distribution and are asymptotically independent from their eigenvectors.

Contribution

It provides the first analysis of the extremal process of principal minors in GOE, including their limiting distribution and joint behavior with eigenvectors.

Findings

Maxima of principal minors follow Gumbel distribution

Maxima are asymptotically independent from eigenvectors

Joint distribution of maxima and eigenvectors derived

Abstract

In this paper, we study the extremal process of the maxima of all the largest eigenvalues of principal minors of the classical Gaussian orthogonal ensemble (GOE). We prove that the fluctuation of the maxima is given by the Gumbel distribution in the limit. We also derive the limiting joint distribution of the maxima and the corresponding eigenvector, which implies that these two random variables are asymptotically independent.