Universality for the minimum modulus of random trigonometric polynomials

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. For discrete distributions we must now deal with possible arithmetic structure in the polynomial evaluated at different points of the circle. On "minor arcs" we obtain strong comparisons with the Gaussian model by translating to a random walk in a high dimensional phase space, and obtaining strong decay estimates on characteristic functions, while major arcs can be handled with cruder arguments. Based on joint work with Hoi Nguyen.

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

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