Abstract
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.
Highlights
This paper concerns the nature of the pairing between the critical points and roots of random polynomials in a single complex variable
Pemantle and Rivin conjectured that when X1, . . . , Xn are chosen to be independent and identically distributed with distribution μ, the empirical distribution constructed from the critical points of pn converges weakly in probability to μ
Theorem 2.3 below gives a bound on the Wasserstein distance between the empirical measures constructed from the roots X1, . . . , Xn and the critical points w1(n), . . . , wn(n−)1 of the polynomial pn defined in (1.1)
Summary
This paper concerns the nature of the pairing between the critical points and roots of random polynomials in a single complex variable. Steinerberger showed that the pairing phenomenon holds for some classes of deterministic polynomials [29], and Kabluchko and Seidel determined the asymptotic fluctuations of the critical point of pn that is nearest a given root [18]. We begin by exhibiting a bound on the Wasserstein, or “transport,” distance between the collections of roots and critical points of pn While this result explains the nearly 1-1 pairing between roots and critical points in Figures 1 and 2, it does not allow one to describe the behavior near any particular root. We accomplish this feat of the paper, where we discuss the joint fluctuations for a fixed number of critical points of pn. Many of our results focus on the cases where the roots X1, . . . , Xn of pn are iid, but for some of our results, we do not even require that the roots be independent
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