Asymptotics of the Number of Components of Random Polynomial Lemniscates

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 ${\mathrm{\Lambda }}_n=\{z\in \mathbb{C}:\mathrm{\mid }{P}_n(z)\mathrm{\mid }<1\}$ be the lemniscate of $P_n$, and let $\mathcal{C}({\mathrm{\Lambda }}_n)$ be the number of connected components of ${\mathrm{\Lambda }}_n$. In this paper, we prove that ${\lim }_{n\to \mathrm{\infty }}\frac{\mathbb{E}[\mathcal{C}({\mathrm{\Lambda }}_n)]}{\sqrt{n}}=\gamma$, and identify the constant $\gamma$.