# Correlations and Pairing Between Zeros and Critical Points of Gaussian
Random Polynomials

Research paper by **Boris Hanin**

Indexed on: **29 Aug '12**Published on: **29 Aug '12**Published in: **Mathematics - Probability**

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#### Abstract

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.