Correlations and pairing between zeros and critical points of random polynomials

Abstract

This dissertation concerns the relationship between zeros and critical points of random polynomials in one complex variable. More precisely, we compute the high degree asymptotics of the two-point function between zeros and critical points for so-called Hermitian Gaussian random polynomials. We prove that for these ensembles, zeros and critical points typically come in pairs and present results that quantitatively describe this pairing.