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

TL;DR

This paper demonstrates that Haar unitaries, when represented in certain high-dimensional irreducible representations, exhibit strong asymptotic freeness almost surely, extending previous results to larger partition sizes.

Contribution

It extends the known regime of strong asymptotic freeness of Haar unitaries to representations arising from partitions of size up to a power of the dimension, surpassing prior logarithmic bounds.

Findings

Strong asymptotic freeness holds for partitions of size up to $n^{1/42- ext{small}}$.

The result is uniform in the specified regime.

It generalizes previous results limited to logarithmic partition sizes.

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 $n^{\frac{1}{42}-\varepsilon}$ 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.