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

Michael Magee; Mikael de la Salle

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arXiv · 2409.03626

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

Michael Magee·Mikael de la Salle

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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.

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BibTeX RIS

Michael Magee, Mikael de la Salle. 2024-09-05. Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations. https://arxiv.org/abs/2409.03626

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