An analytic approach to the finite R-transform
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914530407219200 |
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| author | Arizmendi, Octavio Fujie, Katsunori |
| author_facet | Arizmendi, Octavio Fujie, Katsunori |
| contents | We revisit Marcus' finite free analogue of Voiculescu $R$-transform from an analytic viewpoint. By relating the finite free Fourier transform to the Laplace transform, we study the finite $R$-transform through logarithmic potentials and Legendre transforms. Under suitable assumptions, we prove that the finite $R$-transform of a polynomial differs from the Voiculescu $R$-transform of its empirical root distribution by $O(N^{-1})$. As an application, we obtain an analytic proof of the convergence of finite free additive convolution to free additive convolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02093 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An analytic approach to the finite R-transform Arizmendi, Octavio Fujie, Katsunori Probability Operator Algebras 46L54 We revisit Marcus' finite free analogue of Voiculescu $R$-transform from an analytic viewpoint. By relating the finite free Fourier transform to the Laplace transform, we study the finite $R$-transform through logarithmic potentials and Legendre transforms. Under suitable assumptions, we prove that the finite $R$-transform of a polynomial differs from the Voiculescu $R$-transform of its empirical root distribution by $O(N^{-1})$. As an application, we obtain an analytic proof of the convergence of finite free additive convolution to free additive convolution. |
| title | An analytic approach to the finite R-transform |
| topic | Probability Operator Algebras 46L54 |
| url | https://arxiv.org/abs/2605.02093 |