Asymptotic Results for Random Polynomials on the Unit Circle

2012/11/16 by Gabriel H. Tucci, Tucci, Gabriel H., Philip Whiting +2
Mathematics · Physics and Astronomy · #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR) #Quantum chaos and dynamical systems #math.PR

paper · pdf · doi:10.48550/arxiv.1211.3958

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

arxiv created 2012/11/16 · openalex publication_date 2012/11/16 · arxiv updated 2012/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞ be an infinite sequence of positive integers and let \z_k\_k=1^∞ 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) = ∏_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^∞, the log maximum magnitude of these polynomials scales as s_NI^* where s_N²=∑_k=1^Nn_k² and I^* is a strictly positive random variable.

Citations

• Asymptotic Behavior of the Maximum and Minimum Singular Value of Random Vandermonde Matrices

Related

• Real roots near the unit circle of random polynomials
• Random polynomials: the closest roots to the unit circle
• On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients
• On random polynomials with an intermediate number of real roots
• Critical points of random polynomials with independent identically distributed roots