Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces
Abstract
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 $1O(N^{3/2+3\ep}).$ We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.
 Publication:

arXiv eprints
 Pub Date:
 May 2013
 arXiv:
 arXiv:1305.6105
 Bibcode:
 2013arXiv1305.6105H
 Keywords:

 Mathematics  Complex Variables;
 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 20 pages, 2 figures