Nearest zero-critical point distances for Gaussian SU(2) polynomials

Abstract

We study nearest zero--critical point distances for the Gaussian SU(2) polynomial $p_n$. For each zero $z_i$ of $p_n$, let $D_n(z_i)$ denote its Fubini--Study distance to the nearest critical point. We prove that the empirical measure of the rescaled distances $nD_n(z_i)$ converges weakly in probability to the law of $2/|Z|$, where $Z$ is distributed according to the Fubini--Study probability measure in the affine chart.