Asymptotic Results for Random Polynomials on the Unit Circle

TL;DR

This paper investigates the asymptotic behavior of the maximum magnitude of complex random polynomials with roots uniformly distributed on the unit circle, revealing a scaling law involving the sum of squared multiplicities.

Contribution

It establishes the asymptotic distribution of the maximum magnitude for a broad class of random polynomials with roots on the unit circle, under regularity conditions.

Findings

Log maximum magnitude scales as s_N * I^*

I^* is a strictly positive random variable

Scaling involves the sum of squared multiplicities s_N^2

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 $s_{N}I^{*}$ where $s_{N}^{2}=\sum_{k=1}^{N}{n_{k}^{2}}$ and $I^{*}$ is a strictly positive random variable.