Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations

Abstract

We prove almost sure strong asymptotic freeness of i.i.d. random unitaries with the following law: sample a Haar unitary matrix of dimension $n$ and then send this unitary into an irreducible representation of $U(n)$. The strong convergence holds as long as the irreducible representation arises from a pair of partitions of total size at most $n^{\frac{1}{42}-\epsilon }$ and is uniform in this regime. Previously this was known for partitions of total size up to $\asymp \log n/\log \log n$ by a result of Bordenave and Collins.

54 pages; v2: improvement in the exposition and correction in the exponents