Concentration of Haar measures , with an application to random matrices

We present a novel approach to measure concentration that works through a deeper investigation of the semigroup method. In particular, we show how couplings and rates of convergence of Markov chains can be used to obtain concentration bounds. As an application, we obtain a measure concentration result for random unitary matrices and other kinds of Haar-distributed random variables, which allows us to directly establish the concentration of the empirical distribution of eigenvalues of sums of random hermi-tian matrices. This also gives an example of concentration for discontinuous functions, which can be a significant area of application of the new technique.