On the number of components of random polynomial lemniscates

A lemniscate of a complex polynomial $Q_n$ of degree $n$ is a sublevel set of its modulus, i.e., of the form $\{z\in \mathbb{C}:\mathrm{\mid }{Q}_n(z)\mathrm{\mid }<t\}$ for some $t>0.$ In general, the number of connected components of this lemniscate can vary anywhere between 1 and $n$. In this paper, we study the expected number of connected components for two models of random lemniscates. First, we show that lemniscates whose defining polynomial has i.i.d. roots chosen uniformly from $\mathbb{D}$, has on average $\mathcal{O}(\sqrt{n})$ number of connected components. On the other hand, if the i.i.d. roots are chosen uniformly from ${\mathbb{S}}^1$, we show that the expected number of connected components, divided by n, converges to $\frac{1}{2}$.