[Paper Review] On Zeroes of Random Polynomials and Applications to Unwinding

Stefan Steinerberger, Hau‐Tieng Wu|arXiv (Cornell University)|Jul 15, 2018

Mathematical Dynamics and Fractals17 references4 citations

TL;DR

This paper studies the asymptotic distribution of solutions to the equation $ p_n(z) = p_n(0) $, where $ p_n(z) $ is a random polynomial with roots drawn i.i.d. from a compactly supported, continuous probability measure $ \mu $ on $ \mathbb{C} $. It shows that as $ n \to \infty $, the roots of $ p_n(z) - p_n(0) = 0 $ converge in distribution to a measure $ \nu $ that reproduces $ \mu $ on the set $ A $ where the logarithmic potential exceeds its value at the origin, and concentrates on a curve set $ B $ where the potential is equal, with the density on $ B $ determined by the angular variation of the potential.

ABSTRACT

Let $μ$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots, z_n$ be independent random variables, $z_i \sim μ$, and consider the random polynomial $$ p_n(z) = \prod_{k=1}^{n}{(z - z_k)}.$$ We determine the asymptotic distribution of $\left\{z \in \mathbb{C}: p_n(z) = p_n(0) ight\}$. In particular, if $μ$ is radial around the origin, then those solutions are also distributed according to $μ$ as $n ightarrow \infty$. Generally, the distribution of the solutions will reproduce parts of $μ$ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.

Motivation & Objective

• To understand the limiting distribution of solutions to $ p_n(z) = p_n(0) $ for random polynomials with i.i.d. roots drawn from a compactly supported, continuous probability measure $ \mu $ on $ \mathbb{C} $.
• To characterize the asymptotic behavior of these solutions in terms of potential theory, particularly the logarithmic potential of $ \mu $.
• To extend known results on critical points of random polynomials to the case of level sets $ p_n(z) = p_n(0) $, especially when $ \mu $ is not radial.
• To provide a precise description of the measure $ \nu $ that arises as the weak limit of the root distribution of $ p_n(z) - p_n(0) $, including the structure on the level set $ B $ where the potential equals its value at the origin.

Proposed method

• Define the sets $ A = \{ z : \int \log|x-z| d\mu(x) > \int \log|x| d\mu(x) \} $ and $ B = \{ z : \text{equality holds} \} $, which partition the complex plane based on the logarithmic potential of $ \mu $.
• Use the law of large numbers for logarithmic potentials: $ \frac{1}{n} \sum_{k=1}^n \log|z - z_k| \to \int \log|z - x| d\mu(x) $ in probability as $ n \to \infty $.
• Apply the deviation principle to show that the level set $ \{ z : \sum_{k=1}^n \log|z - z_k| = \sum_{k=1}^n \log|z_k| \} $ converges uniformly to the set $ B $, away from the support of $ \mu $.
• Analyze the argument of the polynomial $ p_n(z) $ along the limiting curve $ C \subset B $, showing that the total variation of $ \arg p_n(z) $ over $ C $ is asymptotically linear in $ n $, leading to a regular distribution of solutions.
• Use the implicit function theorem to show that $ B $ is a $ C^1 $ curve locally, and derive the density on $ B $ as $ \frac{1}{2\pi} \left| \frac{\partial}{\partial t} \int \arg(\gamma(t) - z) d\mu(z) \right|^{-1} $, where $ \gamma $ is an arclength parametrization of $ C $.
• Verify the results via explicit computation for a union of two unit circles, showing that the limiting measure $ \nu $ reproduces $ \mu $ on $ A $ and concentrates on four new curves in $ B $, with numerical simulations confirming the prediction.

Experimental results

Research questions

• RQ1 How does the distribution of solutions to $ p_n(z) = p_n(0) $ behave as $ n \to \infty $ for random polynomials with i.i.d. roots from a compactly supported, continuous measure $ \mu $?
• RQ2 Under what conditions does the solution set reproduce the original measure $ \mu $, and when does it instead concentrate on a curve?
• RQ3 What is the precise structure of the limiting measure $ \nu $ on the set $ B $, where the logarithmic potential of $ \mu $ equals its value at the origin?
• RQ4 How does the angular variation of the potential along $ B $ affect the density of solutions on the limiting curve?
• RQ5 Can the Blaschke unwinding series be analyzed using the asymptotic distribution of roots of $ p_n(z) = p_n(0) $?

Key findings

• For radial $ \mu $, the solutions to $ p_n(z) = p_n(0) $ converge in distribution to $ \mu $ as $ n \to \infty $, meaning the roots of $ p_n(z) - p_n(0) $ reproduce the original measure.
• For general non-radial $ \mu $, the limiting measure $ \nu $ equals $ \mu $ on the set $ A $, where the logarithmic potential exceeds its value at the origin, and has total mass $ 1 - \mu(A) $ supported on the level set $ B $.
• On the set $ B $, the density of solutions is given by $ \frac{1}{2\pi} \left| \frac{\partial}{\partial t} \int \arg(\gamma(t) - z) d\mu(z) \right|^{-1} $, where $ \gamma $ is an arclength parametrization of a connected component of $ B $, under regularity assumptions.
• Numerical simulations confirm that for $ \mu $ supported on two unit circles centered at 0 and 2, the solutions distribute according to $ \mu $ on $ A $ and concentrate on four curves in $ B $, with slight repulsion near the origin due to the root at 0.
• The origin is always a root of $ p_n(z) - p_n(0) $, and a repulsion effect creates a shrinking bubble around 0 with no roots, which vanishes as $ n \to \infty $.
• The expected logarithmic growth of $ |p_n(0)| $ is $ \frac{1}{2} \log 2 $, and the set of points where the logarithmic potential equals this value defines the limiting support of the solutions.