Cutoff profiles: from transpositions to more general graphs
Using Fourier analysis on groups (representation theory) to study card mixing was first done by Diaconis and Shahshahani in their landmark paper on transpositions, and has proved to be a very fruitful technique. We will explain how to improve their L^2-bounds to get second order terms, namely cutoff profiles, and show how this leads to the profile for transpositions and some conjugacy classes. We will then turn our attention to more general graphs, presenting a family of graphs for which the mixing behaviour is very easy to tune, and prove that any cutoff profile is possible.