On the Pairing Between Zeros and Critical Points of Random Polynomials with Independent Roots

Williams, Noah Nelson

Graduate Thesis Or Dissertation

On the Pairing Between Zeros and Critical Points of Random Polynomials with Independent Roots Pubblico Deposited

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/db78tc078

Abstract

Consider the random, complex polynomial p_n(z) = ∏^{n _j}₌₁(z−X_j ), whose roots X₁,..., X_n are complex-valued random variables. It is known that for large n, when the roots are independently and identically distributed (iid), the critical points and roots of p_n are stochastically similar. In particular, Pemantle and Rivin, Kabluchko, Reddy, and others showed that when X₁, X₂,... are iid with distribution μ, then the empirical measure constructed from the critical points of p_n converges to μ in probability as the degree n tends to infinity.

Simulations show that, in fact, the roots and critical points of p_n "pair-up" with one another in a nearly one-to-one fashion, a phenomenon which has been initially investigated by Hanin, O'Rourke, Kabluchko and Seidel, the author, and others. This thesis seeks to quantify root-and-critical-point pairing on several scales, including macroscopic comparisons between entire collections of roots and critical points, microscopic examinations of individual critical points that lie near fixed roots, and a "mesoscopic" local law to explain the situation at scales in between.

In Chapter 2, we show that for a deterministic point ξ lying outside the support of μ, almost surely the polynomial q_n(z) := p_n(z)(z - ξ) has a critical point at distance O(1/n) from ξ. In other words, conditioning the random polynomials p_n to have a root at ξ almost surely forces a critical point near ξ. More generally, we prove an analogous result for the critical points of q_n(z) := p_n(z)(z - ξ₁)...(z - ξ_k), where ξ₁,..., ξ_k are deterministic. In addition, when k = o(n), we show that the empirical distribution constructed from the critical points of q_n converges to μ in probability as the degree tends to infinity, extending a result of Kabluchko.

In Chapter 3, under a regularity assumption, we show that if the roots of p_n are iid, 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/n² 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.

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