The arc length of a random lemniscate

Abstract

A polynomial lemniscate is a curve in the complex plane defined by $\{z\in \mathbb{C}:|p(z)|=t\}$. Erdös, Herzog, and Piranian posed the extremal problem of determining the maximum length of a lemniscate $\Lambda =\{z\in \mathbb{C}:|p(z)|=1\}$ when $p$ is a monic polynomial of degree $n$. In this paper, we study the length and topology of a random lemniscate whose defining polynomial has independent Gaussian coefficients. In the special case of the Kac ensemble we show that the length approaches a nonzero constant as $n\to \infty$. We also show that the average number of connected components is asymptotically $n$, and we observe a positive probability (independent of $n$) of a giant component occurring.

19 pages, 7 figures. This version includes results on the connected components of the lemniscate

References in corpus (5)

• Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
• On the number of connected components of random algebraic hypersurfaces
• Equilibrium distribution of zeros of random polynomials
• On the geometry of random lemniscates
• New estimates for the length of the Erdos-Herzog-Piranian lemniscate