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- W2548614631 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 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 $n rightarrow infty$. 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." @default.
- W2548614631 created "2016-11-11" @default.
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- W2548614631 date "2016-10-31" @default.
- W2548614631 modified "2023-09-27" @default.
- W2548614631 title "The arclength of a random lemniscate in the plane" @default.
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- W2548614631 hasPublicationYear "2016" @default.
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