The asymptotic mixing analysis of random walk on a group
became popular in the 80's and is a growing field, with applications
to cryptography, statistical physics and computer algebra systems. I
will discuss several recent results resolving models which have been
open since the 80's including a new local limit theorem on nilpotent
groups, the asymptotic mixing of the abelian sandpile model on a
square grid and the randomization of a 15-puzzle. The methods use
Fourier techniques, including van der Corput's inequality and the
Gowers-Cauchy-Schwarz inequality.