Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Pairing of zeros and critical points for random polynomials

Boris Hanin

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Abstract

Let $p_{N}$ be a random degree $N$ polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure $\mu$ on the Riemann sphere $S^{2}$. This article proves that if we condition $p_{N}$ to have a zero at some fixed point $\xi\in S^{2}$, then, with high probability, there will be a critical point $w_{\xi}$ at a distance $N^{-1}$ away from $\xi$. This $N^{-1}$ distance is much smaller than the $N^{-1/2}$ typical spacing between nearest neighbors for $N$ i.i.d. points on $S^{2}$. Moreover, with the same high probability, the argument of $w_{\xi}$ relative to $\xi$ is a deterministic function of $\mu$ plus fluctuations on the order of $N^{-1}$.

Résumé

Soit $p_{N}$ un polynôme aléatoire de degré $N$ en une variable complexe tel que ses zéros sont distribués indépendamment suivant une mesure de probabilité $\mu$ fixée et définie sur la sphère de Riemann $S^{2}$. Cet article prouve que si nous conditionnons $p_{N}$ pour avoir un zéro en un point fixé $\xi\in S^{2}$, alors, avec grande probabilité, il y aura un point critique $w_{\xi}$ à une distance $N^{-1}$ de $\xi$. Cette distance $N^{-1}$ est beaucoup plus petite que l’espacement typique entre deux points voisins pour $N$ points i.i.d. sur $S^{2}$, qui lui est d’ordre $N^{-1/2}$. De plus, avec la méme grande probabilité, l’argument de $w_{\xi}$ relativement à $\xi$ est une fonction déterministe de $\mu$, plus des fluctuations d’ordre $N^{-1}$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1498-1511.

Dates
Received: 3 February 2016
Revised: 12 April 2016
Accepted: 22 May 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624049

Digital Object Identifier
doi:10.1214/16-AIHP767

Mathematical Reviews number (MathSciNet)
MR3689975

Zentralblatt MATH identifier
1373.30006

Subjects
Primary: 30C10: Polynomials 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} 60G60: Random fields

Keywords
Zeros Critical points Random polynomials Gauss–Lucas

Citation

Hanin, Boris. Pairing of zeros and critical points for random polynomials. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1498--1511. doi:10.1214/16-AIHP767. https://projecteuclid.org/euclid.aihp/1500624049


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