Pairing of zeros and critical points for random polynomials

Let p(N) 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 S-2. This article proves that if we condition p(N) to have a zero at some fixed point xi is an element of S-2, then, with high probability, there will be a critical point omega xi at a distance N-1 away from xi. This N-1 distance is much smaller than the N-1/2 typical spacing between nearest neighbors for N i.i.d. points on S-2. Moreover, with the same high probability, the argument of omega xi relative to xi is a deterministic function of mu plus fluctuations on the order of N-1.