Asymptotic $\ast$--distribution of permuted Haar unitary matrices

TL;DR

This paper investigates the asymptotic behavior of permuted Haar unitary matrices, establishing conditions under which they become circular and free from unpermuted matrices, with results holding almost surely for random permutations.

Contribution

It identifies specific conditions on permutation sequences that ensure permuted Haar unitaries are asymptotically circular and free, extending understanding of their asymptotic distribution.

Findings

Conditions for asymptotic circularity and freeness are established.

Almost sure convergence for sequences of random permutations.

Results are generic and hold for a broad class of permutations.

Abstract

We study Haar unitary random matrices with permuted entries. For a sequence of permutations $\left(\sigma_N\right)_N$, where $\sigma_N$ acts on $N\times N$ matrices we identify conditions under which the $\ast$--distribution of permuted Haar unitary matrices $U_N^{\sigma_N}$ is asymptotically circular and free from the unpermuted sequence $U_N$. We show that this convergence takes place in the almost sure sense. Moreover we show that our conditions on the sequence of permutations are generic in the sense that are almost surely satisfied by a sequence of random permutations.