  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.
