A simple proof of local universality for roots of Kac polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915948968017920 |
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| author | Michelen, Marcus Yakir, Oren |
| author_facet | Michelen, Marcus Yakir, Oren |
| contents | Let $f_n$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_n$ cluster uniformly around the unit circle as $n$ grows large. We give a simple and self-contained proof of local universality for the correlation functions of the roots at the microscopic scale $1/n$ around a fixed point on the circle. While previous proofs of local universality were focused on studying the logarithmic potential of $f_n$, we instead directly compare the scaled random polynomial to a limiting Gaussian analytic function, and establish convergence of correlations via a soft argument, using only basic complex analysis and an anti-concentration bound of Esseen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A simple proof of local universality for roots of Kac polynomials Michelen, Marcus Yakir, Oren Probability Complex Variables Let $f_n$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_n$ cluster uniformly around the unit circle as $n$ grows large. We give a simple and self-contained proof of local universality for the correlation functions of the roots at the microscopic scale $1/n$ around a fixed point on the circle. While previous proofs of local universality were focused on studying the logarithmic potential of $f_n$, we instead directly compare the scaled random polynomial to a limiting Gaussian analytic function, and establish convergence of correlations via a soft argument, using only basic complex analysis and an anti-concentration bound of Esseen. |
| title | A simple proof of local universality for roots of Kac polynomials |
| topic | Probability Complex Variables |
| url | https://arxiv.org/abs/2511.21455 |