An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929500268265472 |
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| author | O'Rourke, Sean Williams, Noah |
| author_facet | O'Rourke, Sean Williams, Noah |
| contents | If $p:\mathbb{C} \to \mathbb{C}$ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when $p$ is a random polynomial of degree $n$ with roots chosen independently from a radially symmetric, compactly supported probably measure $μ$ in the complex plane. We show that the largest (in magnitude) critical points are closely paired with the largest roots of $p$. This allows us to compute the asymptotic fluctuations of the largest critical points as the degree $n$ tends to infinity. We show that the limiting distribution of the fluctuations is described by either a Gaussian distribution or a heavy-tailed stable distribution, depending on the behavior of $μ$ near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09538 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots O'Rourke, Sean Williams, Noah Probability If $p:\mathbb{C} \to \mathbb{C}$ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when $p$ is a random polynomial of degree $n$ with roots chosen independently from a radially symmetric, compactly supported probably measure $μ$ in the complex plane. We show that the largest (in magnitude) critical points are closely paired with the largest roots of $p$. This allows us to compute the asymptotic fluctuations of the largest critical points as the degree $n$ tends to infinity. We show that the limiting distribution of the fluctuations is described by either a Gaussian distribution or a heavy-tailed stable distribution, depending on the behavior of $μ$ near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials. |
| title | An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots |
| topic | Probability |
| url | https://arxiv.org/abs/2409.09538 |