Universality of the minimum modulus for random trigonometric polynomials

It has been shown in [YZ] 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.

Duke Scholars

Author Nicholas A Cook Mathematics