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

Abstract

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables ${\xi }_0,{\xi }_1,\dots ,{\xi }_n$ 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 $Q_n^{\prime }$. 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, $f(z)=\mathbb{E}[1/(z-{\xi }_k)]$ 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.

28 pages, 4 figures

References in corpus (5)

• Asymptotic distribution of complex zeros of random analytic functions
• Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces
• Zeros of random polynomials and its higher derivatives
• Pairing between zeros and critical points of random polynomials with independent roots
• The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives