# Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces

@article{Hanin2015PairingOZ,
title={Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces},
author={B. Hanin},
journal={Mathematical Research Letters},
year={2015},
volume={22},
pages={111-140}
}
• B. Hanin
• Published 2015
• Mathematics, Physics
• Mathematical Research Letters
• We prove that zeros and critical points of a random polynomial $p_N$ of degree $N$ in one complex variable appear in pairs. More precisely, if $p_N$ is conditioned to have $p_N(\xi)=0$ for a fixed $\xi \in \C\backslash\set{0},$ we prove that there is a unique critical point z in the annulus $N^{-1-\ep}<\abs{z-\xi}< N^{-1+\ep}}$ and no critical points closer to $\xi$ with probability at least $1-O(N^{-3/2+3\ep}).$ We also prove an analogous statement in the more general setting of random… CONTINUE READING
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