Asymptotic root distribution of polynomials under repeated polar differentiation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912554805100544 |
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| author | Perales, Daniel Yang, Zhiyuan |
| author_facet | Perales, Daniel Yang, Zhiyuan |
| contents | Given a sequence of real rooted polynomials $\{p_n\}_{n\geq 1}$ with a fixed asymptotic root distribution, we study the asymptotic root distribution of the repeated polar derivatives of this sequence. This limiting distribution can be seen as the result of fractional free convolution and pushforward maps along Möbius transforms for distributions. This new family of operations on measures forms a semigroup and satisfy some other nice properties. Using the fact that polar derivatives commute with one another, we obtain a non-trivial commutation relation between these new operations. We also study a notion of polar free infinite divisibility and construct Belinschi-Nica type semigroups. Finally, we provide some interesting examples of distributions that behave nicely with respect to these new operations, including the Marchenko-Pastur and the Cauchy distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic root distribution of polynomials under repeated polar differentiation Perales, Daniel Yang, Zhiyuan Probability Combinatorics Operator Algebras 46L54, 26C10 Given a sequence of real rooted polynomials $\{p_n\}_{n\geq 1}$ with a fixed asymptotic root distribution, we study the asymptotic root distribution of the repeated polar derivatives of this sequence. This limiting distribution can be seen as the result of fractional free convolution and pushforward maps along Möbius transforms for distributions. This new family of operations on measures forms a semigroup and satisfy some other nice properties. Using the fact that polar derivatives commute with one another, we obtain a non-trivial commutation relation between these new operations. We also study a notion of polar free infinite divisibility and construct Belinschi-Nica type semigroups. Finally, we provide some interesting examples of distributions that behave nicely with respect to these new operations, including the Marchenko-Pastur and the Cauchy distributions. |
| title | Asymptotic root distribution of polynomials under repeated polar differentiation |
| topic | Probability Combinatorics Operator Algebras 46L54, 26C10 |
| url | https://arxiv.org/abs/2508.18575 |