Probability and Statistics Seminar: Universality for the minimum modulus of random trigonometric polynomials

Friday, February 19 at 3:30pm to 4:30pm

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Nick Cook, Duke University

Abstract: We consider the restriction to the unit circle of random degree-n polynomials with iid normalized coefficients (Kac polynomials). Recent work of Yakir and Zeitouni shows that for Gaussian coefficients, the minimum modulus (suitably rescaled) follows a limiting exponential distribution. We show this is a universal phenomenon, extending their result to arbitrary sub-Gaussian coefficients, such as Rademacher signs. Our approach relates the joint distribution of small values at several angles to that of a random walk in high-dimensional phase space, for which we obtain strong central limit theorems. For discrete coefficients this requires dealing with arithmetic structure among the angles. Based on joint work with Hoi Nguyen.

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