Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917939815383040 |
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| author | Magee, Michael de la Salle, Mikael |
| author_facet | Magee, Michael de la Salle, Mikael |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_03626 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations Magee, Michael de la Salle, Mikael Probability Group Theory Operator Algebras Representation Theory 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. |
| title | Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations |
| topic | Probability Group Theory Operator Algebras Representation Theory |
| url | https://arxiv.org/abs/2409.03626 |