Electronic Journal of Probability

Distances between zeroes and critical points for random polynomials with i.i.d. zeroes

Zakhar Kabluchko and Hauke Seidel

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Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi _0,\xi _1,\ldots ,\xi _n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to \infty $, with high probability there is a critical point of $Q_n$ which is very close to $\xi _0$. We localize the position of this critical point by proving that the difference between $\xi _0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\xi _0))$ and variance of order $\log n \cdot n^{-3}$. Here, $f(z)= \mathbb E [\frac 1 {z-\xi _k}]$ is the Cauchy–Stieltjes transform of the $\xi _k$’s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 34, 25 pp.

Received: 5 July 2018
Accepted: 17 March 2019
First available in Project Euclid: 9 April 2019

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Digital Object Identifier

Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
Secondary: 60G57: Random measures 60B10: Convergence of probability measures

random polynomials critical points i.i.d. zeroes non-normal domain of attraction of the normal law functional limit theorems random analytic functions

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Kabluchko, Zakhar; Seidel, Hauke. Distances between zeroes and critical points for random polynomials with i.i.d. zeroes. Electron. J. Probab. 24 (2019), paper no. 34, 25 pp. doi:10.1214/19-EJP295. https://projecteuclid.org/euclid.ejp/1554775413

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