Correlations and pairing between zeros and critical points of Gaussian Random polynomials

Let p_N be a degree N random polynomial in a complex variable. We obtain an explicit asymptotic formula for the covariance between the counting measures of its zeros and critical points, which we denote Cov_N(z,w). This formula shows that the correlation between a zero at z and a critical point at w is short range, decaying like e^-N|z-w|2. With |z-w| on the order of N^-1/2, however, Cov_N(z,w) is sharply peaked near z= w, causing zeros and critical points to appear in pairs. We prove bounds on the expected distance and angular dependence between a critical point and its paired zero.