Asymptotic results for random polynomials on the unit circle

Abstrakt

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 {n_k}_k=1^∞ be an infinite sequence of positive integers and  let {z_k}_k=1^∞ 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 P_N^z =Π^N_k=1 z − z_k^nk 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 {n_k}^∞_k=1, the log maximum magnitude of these polynomials scales as s_NI, where s²_N =Σ^N_k=1 n_k² and I is a strictly positive random variable.