Pairing of zeros and critical points for random meromorphic functions on riemann surfaces

We prove that zeros and critical points of a random polynomial pN of degree N in one complex variable appear in pairs. More precisely, suppose pN is conditioned to have pN(ξ)=0 for a fixed ξεC. For εε (0, 1/ 2) we prove that there is a unique critical point in the annulus {z ∈ C/N^-1-ε</z -ξ/< N^-1+ε}and no critical points closer to ξ with probability at least 1-O(N^-3/2+3ε). We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.