Some recent methods for establishing universality limits Original Research Article

We survey some recent methods for establishing universality limits for random matrices in the unitary case. These include Levin’s method using a Markov–Bernstein inequality, a comparison inequality of the author, and a method based on complex analysis and reproducing kernels. We focus on the bulk of the spectrum for measures with compact support, but the methods may also be used at the soft or hard edge, and for measures with unbounded support.