On the number of components of random polynomial lemniscates

Abstract

A lemniscate of a complex polynomial $Q_n$ of degree n is a sublevel set of its modulus, i.e., of the form $\{z\in \mathbb{C}:|Q_n(z)|<t\}$ for some $t>0$. In general, the number of connected components of a unit lemniscate (i.e. for $t=1$) 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 the expected number of connected components of lemniscates whose defining polynomial has i.i.d. roots chosen uniformly from $\mathbb{D}$, is bounded above and below by a constant multiple $\sqrt{n}$. 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 $\frac{1}{2}$.