Asymptotics of the Number of Components of Random Polynomial Lemniscates
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910268808757248 |
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| author | Ghosh, Subhajit Ramachandran, Koushik Shekhar, Atul |
| author_facet | Ghosh, Subhajit Ramachandran, Koushik Shekhar, Atul |
| contents | Consider a sequence of random polynomials $P_n(z) = \prod_{k=1}^{n}(z - X_k)$, where $\{X_k\}_k$ are i.i.d. random variables distributed uniformly on the unit disc $\mathbb{D}$. Let $Λ_n = \{z \in \mathbb{C}: |P_n(z)| < 1\}$ be the lemniscate of $P_n$, and let $\mathscr{C}(Λ_n)$ be the number of connected components of $Λ_n$. In this paper, we prove that $\lim_{n\to\infty}\frac{\mathbb{E}[\mathscr{C}(Λ_n)]}{\sqrt{n}}= γ$, and identify the constant $γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29095 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the Number of Components of Random Polynomial Lemniscates Ghosh, Subhajit Ramachandran, Koushik Shekhar, Atul Probability Classical Analysis and ODEs Complex Variables 60D05, 30C15 Consider a sequence of random polynomials $P_n(z) = \prod_{k=1}^{n}(z - X_k)$, where $\{X_k\}_k$ are i.i.d. random variables distributed uniformly on the unit disc $\mathbb{D}$. Let $Λ_n = \{z \in \mathbb{C}: |P_n(z)| < 1\}$ be the lemniscate of $P_n$, and let $\mathscr{C}(Λ_n)$ be the number of connected components of $Λ_n$. In this paper, we prove that $\lim_{n\to\infty}\frac{\mathbb{E}[\mathscr{C}(Λ_n)]}{\sqrt{n}}= γ$, and identify the constant $γ$. |
| title | Asymptotics of the Number of Components of Random Polynomial Lemniscates |
| topic | Probability Classical Analysis and ODEs Complex Variables 60D05, 30C15 |
| url | https://arxiv.org/abs/2605.29095 |