Semicircular families of general covariance from Wigner matrices with permuted entries

TL;DR

This paper characterizes the limiting distributions of permuted Wigner matrices, revealing conditions for semicircular families with general covariance and independence properties, based on permutation geometry.

Contribution

It provides a new characterization of the limiting traffic and joint distributions of permuted Wigner matrices using permutation geometry.

Findings

Describes limiting traffic distribution via permutation geometry.

Identifies conditions for traffic and free independence.

Establishes semicircular family structures with general covariance.

Abstract

Let $(\sigma_N^{(i)})_{i \in I}$ be a family of symmetric permutations of the entries of a Wigner matrix $\mathbf{W}_N$. We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices $(\mathbf{W}_N^{\sigma_N^{(i)}})_{i \in I}$ in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of $(\mathbf{W}_N^{\sigma_N^{(i)}})_{i \in I}$. In particular, we obtain a description in terms of semicircular families with general covariance structures. As a special case, we derive necessary and sufficient conditions for traffic independence as well as sufficient conditions for free independence.