The arc length and topology of a random lemniscate

Abstract: A polynomial lemniscate is a curve in the complex plane defined by \({\{z \in\mathbb{C}: |p(z)| = t\}}\) Erdos, Herzog and Piranian posed the extremal problem of finding the maximum length of a lemniscate \([\Lambda = \{z\in \mathbb{C}: |p(z)| = 1\}] \) when [p] is a monic polynomial of degree [n.] We study the length and topology of a random lemniscate whose defining polynomial has Gaussian coefficients. When the polynomial is sampled from the Kac ensemble, we show that the expected length approaches a nonzero constant as \([n\rightarrow\infty,] where [ deg(p) =n.]\) We also prove that the average number of components on a random lemniscate is asymptotically [n] and that there is a positive probability (independent of [n]) of a giant component. This talk is based on joint work with Erik Lundberg.