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Sums of random polynomials with differing degrees

Isabelle Kraus, Marcus Michelen, Sean O'Rourke

Abstract

Let $μ$ and $ν$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$, whose roots are chosen independently from $μ$ and $ν$, respectively. Under assumptions on the measures $μ$ and $ν$, the limiting distribution for the zeros of the sum $p+q$ was by computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021) 124719] as $n \to \infty$. In this paper, we generalize and extend this result to the case where $p$ and $q$ have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of $μ$ and $ν$, scaled by the limiting ratio of the degrees of $p$ and $q$. Additionally, our approach provides a complete description of the limiting distribution for the zeros of $p + q$ for any pair of measures $μ$ and $ν$, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.

Sums of random polynomials with differing degrees

Abstract

Let and be probability measures in the complex plane, and let and be independent random polynomials of degree , whose roots are chosen independently from and , respectively. Under assumptions on the measures and , the limiting distribution for the zeros of the sum was by computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021) 124719] as . In this paper, we generalize and extend this result to the case where and have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of and , scaled by the limiting ratio of the degrees of and . Additionally, our approach provides a complete description of the limiting distribution for the zeros of for any pair of measures and , with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.

Paper Structure

This paper contains 15 sections, 20 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions of this paper
  3. Main results
  4. The light-tailed case
  5. Heavy-tailed case
  6. Outline of the paper
  7. Notation and an overview of the proofs
  8. Notation
  9. Overview of the proof of Theorem \ref{['thm:nolog']}
  10. Proof of Theorem \ref{['thm:main']}
  11. Proof of Lemma \ref{['lem:lpbound']}
  12. Proof of Lemma \ref{['lem:point']}
  13. Proof of Lemma \ref{['lem:tightness']}
  14. Proof of Theorem \ref{['thm:nolog']}
  15. Convergence of random measures

Key Result

Theorem 1.2

Let $m \geq 2$ be a fixed integer, and assume $\mu_1, \ldots, \mu_m \in \mathcal{P}(\mathbb{C})$ have compact support. Assume for each $1 \leq k \leq m-1$, the measure $\mu_k$ is not supported on a circleA measure $\mu \in \mathcal{P}(\mathbb{C})$ is said to be supported on a circle if there exists Then there exists a (deterministic) probability measure $\rho$ on $\mathbb{C}$ so that, for any $\v

Figures (2)

  • Figure 1: A numerical simulation of Example \ref{['ex:main']}. The red squares represent the roots of $p_{n,1}$, which are uniform on the unit disk centered at the origin. In this simulation, $p_{n,1}$ has degree $300$. The blue circles represent the roots of $p_{n,2}$, which are uniform on the unit disk centered at $2$. In this simulation, $p_{n,2}$ has degree $25$. The black crosses represent the roots of the sum $p_{n,1} + p_{n,2}$.
  • Figure 2: A numerical simulation of Theorem \ref{['thm:nolog']}. The red squares represent the roots of $p_n$, which are chosen with respect to a rotationally symmetric distribution that does not have a logarithmic moment. The blue circles are the roots of $q_n$, which are chosen according to the standard complex Gaussian distribution. Both polynomials have degree $200$. The black crosses represent the roots of the sum $p_n + q_n$. The image was cropped and does not display the largest roots (in magnitude) of $p_n$ or $p_n + q_n$.

Theorems & Definitions (42)

  • Definition 1.1
  • Theorem 1.2: Theorem 1.10 in ORourke
  • Definition 1.3: Weak convergence of (random) probability measures
  • Theorem 1.4: Main result: light-tailed case
  • Example 1.5
  • Example 1.6
  • Theorem 1.7: Main result: heavy-tailed case
  • Lemma 2.1: Dominated convergence; Lemma 3.1 from TVcirc
  • Lemma 2.2
  • Lemma 2.3
  • ...and 32 more