The arc length of a random lemniscate

TL;DR

This paper investigates the geometric properties of random polynomial lemniscates, revealing that their length stabilizes to a constant and their topology exhibits predictable asymptotic behavior as the degree increases.

Contribution

It provides the first analysis of the length and topology of random lemniscates with Gaussian coefficients, including asymptotic results for the length and number of components.

Findings

Length approaches a nonzero constant as degree increases

Average number of components asymptotically equals the degree

Positive probability of a giant component occurring

Abstract

A polynomial lemniscate is a curve in the complex plane defined by $\{z \in \mathbb{C}:|p(z)|=t\}$. Erd\"os, Herzog, and Piranian posed the extremal problem of determining the maximum length of a lemniscate $\Lambda=\{ z \in \mathbb{C}:|p(z)|=1\}$ when $p$ is a monic polynomial of degree $n$. In this paper, we study the length and topology of a random lemniscate whose defining polynomial has independent Gaussian coefficients. In the special case of the Kac ensemble we show that the length approaches a nonzero constant as $n \rightarrow \infty$. We also show that the average number of connected components is asymptotically $n$, and we observe a positive probability (independent of $n$) of a giant component occurring.