Almost sure behavior of the critical points of random polynomials
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866929264873439232 |
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| author | Angst, Jürgen Malicet, Dominique Poly, Guillaume |
| author_facet | Angst, Jürgen Malicet, Dominique Poly, Guillaume |
| contents | Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $μ$ and let $P_n(X):=\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\mathbb C[X]$. In [Kab15], the author established the conjecture stated by Pemantle and Rivin in [PR13] that the empirical measure $ν_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $μ$. In this note, we establish that the convergence in fact holds in the almost sure sense. Our result positively answers a question raised by Z. Kabluchko and formalized as a conjecture in the recent paper [MV22]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_06973 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Almost sure behavior of the critical points of random polynomials Angst, Jürgen Malicet, Dominique Poly, Guillaume Probability 30C15, 60G57, 60B10 Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $μ$ and let $P_n(X):=\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\mathbb C[X]$. In [Kab15], the author established the conjecture stated by Pemantle and Rivin in [PR13] that the empirical measure $ν_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $μ$. In this note, we establish that the convergence in fact holds in the almost sure sense. Our result positively answers a question raised by Z. Kabluchko and formalized as a conjecture in the recent paper [MV22]. |
| title | Almost sure behavior of the critical points of random polynomials |
| topic | Probability 30C15, 60G57, 60B10 |
| url | https://arxiv.org/abs/2301.06973 |