Large permutation invariant random matrices are asymptotically free over the diagonal

TL;DR

This paper demonstrates that large permutation invariant random matrices become asymptotically free over the diagonal, extending previous results to broader classes including sparse and variance-profile matrices.

Contribution

It proves asymptotic freeness over the diagonal for permutation invariant matrices under relaxed conditions, including graph-based estimates and entrywise multiplication.

Findings

Asymptotic freeness holds for permutation invariant matrices in probability and expectation.

Results extend to sparse regimes and matrices with variance profiles.

The approach accommodates matrices multiplied by bounded random variables.

Abstract

We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (for example, to Wigner matrices with exploding moments and so the sparse regime of the Erd\H{o}s-R\'{e}nyi model). The result still holds even if the matrices are multiplied entrywise by bounded random variables (for example, as in the case of matrices with a variance profile and percolation models).