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, suppose $p_N$ is conditioned to have $p_N(\xi)=0$ for a fixed $\xi \in \mathbb{C}$. For $\epsilon \in (0, \frac{1}{2})$ we prove that there is a unique critical point in the annulus $\lbrace z \in \mathbb{C} \vert N^{-1-\epsilon} \lt \vert z-\xi \vert \lt N^{-1+\epsilon} \rbrace$ and no critical points closer to $\xi$ with probability at least $1 - O(N^{-3/2+3\epsilon})$. We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.

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Published 13 April 2015