Our starting point concerns a discrete uncertainty principle: how small can the support sets of a vector and its discrete Fourier transform be? By taking a probabilistic and geometric approach we relate this question to the third ensemble of random matrix theory, the Jacobi ensemble. We present the limiting empirical spectral distribution of a random matrix arising in the discrete Fourier setting and the first universality result for the Jacobi ensemble. We discuss the relationship between these two types of random matrices, as well an unexpected instance of universality. This talk is partially based on joint work with Laszlo Erdos.