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

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 $p_N(ξ)=0$ for a fixed $ξ\in \C\backslash\set{0},$ we prove that there is a unique critical point z in the annulus $N^{-1-\ep}<\abs{z-ξ}< N^{-1+\ep}}$ and no critical points closer to $ξ$ 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.

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