Asymptotic Results for Random Polynomials on the Unit Circle

Abstract

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 $\{n_k{\}}_{k=1}^{\infty }$ be an infinite sequence of positive integers and let $\{z_k{\}}_{k=1}^{\infty }$ be a sequence of i.i.d. uniform distributed random variables on the unit circle. The above pair of sequences determine a sequence of random polynomials $P_N(z)={\prod }_{k=1}^N(z-z_k{)}^{n_k}$ 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 $\{n_k{\}}_{k=1}^{\infty }$, the log maximum magnitude of these polynomials scales as $s_N{I}^{\ast }$ where $s_N^2={\sum }_{k=1}^N{n}_k^2$ and $I^{\ast }$ is a strictly positive random variable.

14 pages. arXiv admin note: text overlap with arXiv:1202.3184