  We study the pairing between zeros and critical points of the polynomial
$p_n(z) = \prod_{j=1}^n(z-X_j)$, whose roots $X_1, \ldots, X_n$ are
complex-valued random variables. Under a regularity assumption, we show that if
the roots are independent and identically distributed, the Wasserstein distance
between the empirical distributions of roots and critical points of $p_n$ is on
the order of $1/n$, up to logarithmic corrections. The proof relies on a
careful construction of disjoint random Jordan curves in the complex plane,
which allow us to naturally pair roots and nearby critical points. In addition,
we establish asymptotic expansions to order $1/n^2$ for the locations of the
nearest critical points to several fixed roots. This allows us to describe the
joint limiting fluctuations of the critical points as $n$ tends to infinity,
extending a recent result of Kabluchko and Seidel. Finally, we present a local
law that describes the behavior of the critical points when the roots are
neither independent nor identically distributed.
