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

TL;DR

This paper demonstrates that zeros and critical points of random polynomials and meromorphic functions on Riemann surfaces tend to form pairs, with critical points closely associated with zeros in a probabilistic sense.

Contribution

It establishes a probabilistic pairing between zeros and critical points for random polynomials and meromorphic functions, extending previous results to more general Riemann surface settings.

Findings

Critical points are uniquely paired with zeros within a small annulus.

Probability of finding additional critical points near a zero is negligible.

Results extend to 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 $1-O(N^{-3/2+3\ep}).$ We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.