Universality of the minimum modulus for random trigonometric polynomials

2021/01/18 by Nicholas A. Cook, Cook, Nicholas A., Hoi H. Nguyen +1 · 2 citations
Mathematics · #Geometry and complex manifolds #Analytic Number Theory Research #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2101.07203

Abstract

It has been shown in a recent work by Yakir-Zeitouni that the minimum modulus of random trigonometric polynomials with Gaussian coefficients has a limiting exponential distribution. We show this is a universal phenomenon. Our approach relates the joint distribution of small values of the polynomial at a fixed number m of points on the circle to the distribution of a certain random walk in a 4m-dimensional phase space. Under Diophantine approximation conditions on the angles, we obtain strong small ball estimates and a local central limit theorem for the distribution of the walk.

Cited by

• Law of large numbers for the discriminant of random polynomials
• Limit law for root separation in random polynomials

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