  We present new concentration of measure inequalities for Markov chains,
generalising results for chains that are contracting in Wasserstein distance.
These are particularly suited to establishing the cut-off phenomenon for
suitable chains. We apply our discrete-time inequality to the well-studied
Bernoulli-Laplace model of diffusion, and give a probabilistic proof of
cut-off, recovering and improving the bounds of Diaconis and Shahshahani. We
also extend the notion of cut-off to chains with an infinite state space, and
illustrate this in a second example, of a two-host model of disease in
continuous time. We give a third example, giving concentration results for the
supermarket model, illustrating the full generality and power of our results.
