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On the local pairing behavior of critical points and roots of random polynomials. (English) Zbl 1451.30018

Summary: 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.

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
60F05 Central limit and other weak theorems
60B10 Convergence of probability measures
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