Upgraded free independence phenomena for random unitaries
Abstract
We study upgraded free independence phenomena for unitary elements u 1 u_{1} , u 2 , u_{2}, … representing the large- n n limit of Haar random unitaries, showing that free independence extends to several larger algebras containing u j u_{j} in the ultraproduct of matrices ∏ n → U M n ( C ) \prod _{n \to \mathcal {U}} M_{n}(\mathbb {C}) . Using a uniform asymptotic freeness argument and volumetric analysis, we prove free independence of the Pinsker algebras P j \mathcal {P}_{j} containing u j u_{j} . The Pinsker algebra P j \mathcal {P}_{j} is the maximal subalgebra containing u j u_{j} with vanishing 1 1 -bounded entropy (see Hayes [Int. Math. Res. Not. IMRN 1 (2018), pp. 57–137]); P j \mathcal {P}_{j} in particular contains the relative commutant
Fields of Study & Subjects
Random Matrices and Applications
Geometry and complex manifolds
Probability and Risk Models
Publication details
- Journal / source
- Transactions of the American Mathematical Society Series B
- Publisher
- American Mathematical Society (AMS)
- Volume
- 13
- Issue
- 1
- Pages
- 1–29
- Publication date
- 15 January 2026
- Language
- EN