Asymptotics of expected number of components for random lemniscate with iid roots
For a random polynomial with iid roots in a disc, we consider its level sets (aka lemniscates). We obtain asymptotics for the expected number of connected components. This number shows a phase transition as the radius of the disc approaches one. At the critical value R=1, it behaves like √n. This behaviour can be accredited to a pairing phenomenon between roots and critical points of the given polynomial. This talk is based on an ongoing joint work with Koushik Ramachandran and Subhajit Ghosh.
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