Asymptotic free independence and entry permutations for Gaussian random matrices. Part II: Infinitesimal freeness

TL;DR

This paper investigates the infinitesimal distributional properties of Gaussian Unitary Ensemble matrices with permuted entries, revealing conditions under which they are infinitesimally free or not, especially focusing on the transpose permutation.

Contribution

It extends the theory of asymptotic infinitesimal freeness to permuted GUE matrices, including the case of the transpose, and characterizes when infinitesimal freeness holds.

Findings

Permuted GUE matrices with uniform permutations have null infinitesimal distribution.

Different permutations of the same GUE matrix are infinitesimally free asymptotically.

GUE matrices are not infinitesimally free from their transpose, despite being asymptotically free.

Abstract

We study asymptotic infinitesimal distributions of Gaussian Unitary Ensembles with permuted entries. We show that for random uniform permutations, the asymptotically permuted GUE matrix has a null infinitesimal distribution. Moreover, we show that asymptotically different permutations of the same GUE matrix are infinitesimally free. Besides this we study particular example of entry permutation - the transpose, and we show that while a GUE matrix is asymptotically free from its transpose it is not infinitesimally free from it.