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• W2897524453 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." @default.
• W2897524453 created "2018-10-26" @default.
• W2897524453 creator A5042318286 @default.
• W2897524453 creator A5069134947 @default.
• W2897524453 date "2018-10-16" @default.
• W2897524453 modified "2023-09-27" @default.
• W2897524453 title "On the local pairing behavior of critical points and roots of random polynomials." @default.
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• W2897524453 hasPublicationYear "2018" @default.
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