Asymptotics of the Number of Components of Random Polynomial Lemniscates

Abstract

Consider a sequence of random polynomials $P_n(z)={\prod }_{k=1}^n(z-X_k)$, where $\{X_k{\}}_k$ are i.i.d. random variables distributed uniformly on the unit disc $\mathbb{D}$. Let ${\Lambda }_n=\{z\in \mathbb{C}:|P_n(z)|<1\}$ be the lemniscate of $P_n$, and let $\mathcal{C}({\Lambda }_n)$ be the number of connected components of ${\Lambda }_n$. In this paper, we prove that ${\lim }_{n\to \infty }\frac{\mathbb{E}[\mathcal{C}({\Lambda }_n)]}{\sqrt{n}}=\gamma$, and identify the constant $\gamma$.

31 pages