Codes and Expansions (CodEx) Seminar

David Jekel (University of Copenhagen)
Asymptotic freeness and commutation

Voiculescu’s free probability theory describes the asymptotic behavior of many families of \(n \times n\) random matrices in the large-\(n\) limit, specifically to find the limit of ”joint moments”–the normalized trace of non-commutative polynomials in the matrices \(X_1^{(n)}\), …, \(X_m^{(n)}\). In particular, independent random matrices tend to become freely independent in the limit, a condition on the joint moments that generalizes of the behavior of free products of groups.

Let \(U_1^{(n)}\), \(U_2^{(n)}\), …be independent Haar unitary matrices. We show that if \(B_j^{(n)}\) is any random matrix such that \(U_j^{(n)}\) and \(B_j^{(n)}\) asymptotically commute for each \(j\), then \(B_1^{(n)}\), \(B_2^{(n)}\), …are freely independent. In other words, free independence of the \(U_j^{(n)}\) is also inherited by any family of matrices which individually asymptotically commute with the respective \(U_j^{(n)}\)’s. This result also extends to iterated chains where the consecutive terms asymptotically commute. The proof is based on combining volume and covering-number estimates with concentration of measure.