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Central limit theorems for the real zeros of Weyl polynomials
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 142, Number 5, October 2020
- pp. 1327-1369
- 10.1353/ajm.2020.0034
- Article
- Additional Information
Abstract:
We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=\xi_0+\xi_1 x+\cdots+{1\over\sqrt{n!}}\xi_n x^n$, where $\xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real roots.