Line up a deck of 52 cards on a table. Randomly choose two cards and switch them. How many switches are needed in order to mix up the deck? Starting from a few concrete problems such as random walks on the discrete circle and the finite ultrametric space this book develops the necessary tools for the asymptotic analysis of these processes. This detailed study culminates with the case-by-case analysis of the cut-off phenomenon discovered by Persi Diaconis. This self-contained text is ideal for graduate students and researchers working in the areas of representation theory group theory harmonic analysis and Markov chains. Its topics range from the basic theory needed for students new to this area to advanced topics such as the theory of Green's algebras the complete analysis of the random matchings and the representation theory of the symmetric group. Language: English