Asymptotic results for random polynomials on the unit circle

In this paper we study the asymptotic behavior of the maximum magnitude of a complex random polynomial with i.i.d. uniformly distributed random roots on the unit circle. More specifically, let (formula presented) be an infinite sequence of positive integers and let (formula presented) be a sequence of i.i.d. uniformly distributed random variables on the unit circle. The above pair of sequences determine a sequence of random polynomials (formula presented) with random roots on the unit circle and their corresponding multiplicities. In this work, we show that subject to a certain regularity condition on the sequence (formula presented), the log maximum magnitude of these polynomials scales as sNI^*, where (formula presented) is a strictly positive random variable.