We study the asymptotics of correlations and nearest neighbor spacings between zeros and holomorphic critical points of $p_N$, a degree N Hermitian Gaussian random polynomial in the sense of Shiffman and Zeldtich, as N goes to infinity. By holomorphic critical point we mean a solution to the equation $\frac{ d}{ dz}p_N(z)=0.$ Our principal result is an explicit asymptotic formula for the local scaling limit of $\E{ Z_{ p_N}\wedge C_{ p_N}},$ the expected joint intensity of zeros and critical points, around any point on the Riemann sphere. Here $Z_{ p_N}$ and $C_{ p_N}$ are the currents of integration (i.e. counting measures) over the zeros and critical points of $p_N$, respectively. We prove that correlations between zeros and critical points are short range, decaying like $e^{ -N\abs{ z-w}^2}.$ With $\abs{ z-w}$ on the order of $N^{ -1/2},$ however, $\E{ Z_{ p_N}\wedge C_{ p_N}}(z,w)$ is sharply peaked near $z=w,$ causing zeros and critical points to appear in rigid pairs. We compute tight bounds on the expected distance and angular dependence between a critical point and its paired zero.
Categories: math.PR math-ph math.CV math.MP