Abstract
A lemniscate of a complex polynomial of degree n is a sublevel set of its modulus, i.e., of the form for some . In general, the number of connected components of a unit lemniscate (i.e. for ) 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 , is bounded above and below by a constant multiple . On the other hand, if the i.i.d. roots are chosen uniformly from , we show that the expected number of connected components, divided by n, converges to .
Acknowledgments
The author expresses gratitude to his thesis advisor Dr. Koushik Ramachandran for suggesting the problem and for feedback on the article. The author deeply appreciates the support, encouragement, and numerous simulating conversations he received from his advisor throughout this project. The author is grateful to the anonymous referees for their detailed feedback and corrections, which have improved the exposition of the paper.
Citation
Subhajit Ghosh. "On the number of components of random polynomial lemniscates." Electron. J. Probab. 29 1 - 24, 2024. https://doi.org/10.1214/24-EJP1147
Information