Pairing of Zeros and Critical Points for Random Polynomials

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Let pₙ be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure mu on the Riemann sphere S². This article proves that if we condition pₙ to have a zero at some fixed point xi in, then, with high probability, there will be a critical point wₓi a distance 1/N away from xi. This 1/N distance is much smaller than the one over root N typical spacing between nearest neighbors for N i.i.d. points on S². Moreover, with the same high probability, the argument of wₓi relative to xi is a deterministic function of mu plus fluctuations on the order of 1/N.