A sharp symmetrized free transport-entropy inequality for the semicircular law
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909440760872960 |
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| author | Diez, Charles-Philippe |
| author_facet | Diez, Charles-Philippe |
| contents | In this paper, following the recent work of Fathi (2018) in the classical case, we provide by two different methods a sharp symmetrized free Talagrand inequality for the semicircular law, which improves the free TCI of Biane and Voiculescu (2000). The first proof holds only in the one-dimensional case and has the advantage of providing a connection with the machinery of free moment maps introduced by Bahr and Boschert (2023) and a free reverse Log-Sobolev inequality. This case also and sheds light on a dual formulation via the free version of the functional Blaschke-Santalo inequality. The second proof gives the result in a multidimensional setting and relies on a random matrix approximation approach developed by Biane (2003), Hiai, Petz and Ueda (2004) combined with Fathi's inequality on Euclidean spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02715 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A sharp symmetrized free transport-entropy inequality for the semicircular law Diez, Charles-Philippe Operator Algebras Functional Analysis Probability In this paper, following the recent work of Fathi (2018) in the classical case, we provide by two different methods a sharp symmetrized free Talagrand inequality for the semicircular law, which improves the free TCI of Biane and Voiculescu (2000). The first proof holds only in the one-dimensional case and has the advantage of providing a connection with the machinery of free moment maps introduced by Bahr and Boschert (2023) and a free reverse Log-Sobolev inequality. This case also and sheds light on a dual formulation via the free version of the functional Blaschke-Santalo inequality. The second proof gives the result in a multidimensional setting and relies on a random matrix approximation approach developed by Biane (2003), Hiai, Petz and Ueda (2004) combined with Fathi's inequality on Euclidean spaces. |
| title | A sharp symmetrized free transport-entropy inequality for the semicircular law |
| topic | Operator Algebras Functional Analysis Probability |
| url | https://arxiv.org/abs/2410.02715 |