Universality for the minimum modulus of random trigonometric polynomials

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. This case of discrete coefficients is particularly challenging, as the distribution is then sensitive to arithmetic structure among the angles. Based on joint work with Hoi Nguyen.