Tensorial free convolution, semicircular, free Poisson and R-transform in high order
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916634728333312 |
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| author | Bonnin, Remi |
| author_facet | Bonnin, Remi |
| contents | This work builds on our previous developments regarding a notion of freeness for tensors. We aim to establish a tensorial free convolution for compactly supported measures. First, we define higher-order analogues of the semicircular (or Wigner) law and the free Poisson (or Marcenko-Pastur) law, giving their moments and free cumulants. We prove the convergence of a Wishart-type tensor to the free Poisson law and recall the convergence of a Wigner tensor to the semicircular law. We also present a free Central Limit Theorem in this context. Next, we introduce a tensorial free convolution, define the $R$-transform, and provide the first examples of free convolution of measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02572 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tensorial free convolution, semicircular, free Poisson and R-transform in high order Bonnin, Remi Operator Algebras Mathematical Physics Combinatorics Probability This work builds on our previous developments regarding a notion of freeness for tensors. We aim to establish a tensorial free convolution for compactly supported measures. First, we define higher-order analogues of the semicircular (or Wigner) law and the free Poisson (or Marcenko-Pastur) law, giving their moments and free cumulants. We prove the convergence of a Wishart-type tensor to the free Poisson law and recall the convergence of a Wigner tensor to the semicircular law. We also present a free Central Limit Theorem in this context. Next, we introduce a tensorial free convolution, define the $R$-transform, and provide the first examples of free convolution of measures. |
| title | Tensorial free convolution, semicircular, free Poisson and R-transform in high order |
| topic | Operator Algebras Mathematical Physics Combinatorics Probability |
| url | https://arxiv.org/abs/2412.02572 |