#### Date of Award

Spring 1-1-2019

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### First Advisor

Sean O'Rourke

#### Second Advisor

Janos Englander

#### Third Advisor

Boris Hanin

#### Fourth Advisor

Sergei Kuznetsov

#### Fifth Advisor

Judith A. Packer

#### Abstract

Consider the random, complex polynomial *p _{n}(z)* = [product]

_{{j=1}}

^{n}(

*z*-X

_{j}), whose roots X

_{1},..., 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*are stochastically similar. In particular, Pemantle and Rivin, Kabluchko, Reddy, and others showed that when X

_{n}_{1}, X

_{2},... are iid with distribution μ, then the empirical measure constructed from the critical points of

*p*converges to μ in probability as the degree

_{n}*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*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

_{n}*q*(

_{n}*z*) :=

*p*(

_{n}*z*)(z - ξ

_{1})...(

*z*- ξ

_{k}), where ξ

_{1},..., ξ

_{k}are deterministic. In addition, when

*k*=

*o*(

*n*), we show that the empirical distribution constructed from the critical points of

*q*converges to μ in probability as the degree tends to infinity, extending a result of Kabluchko.

_{n}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*is on the order of 1/

_{n}*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*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

^{2}*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.

#### Recommended Citation

Williams, Noah Nelson, "On the Pairing Between Zeros and Critical Points of Random Polynomials with Independent Roots" (2019). *Mathematics Graduate Theses & Dissertations*. 71.

https://scholar.colorado.edu/math_gradetds/71