Title: Mixing for colorings, independent sets, random walks: sharp
transition or steady convergence?

Abstract: MCMC samplers are commonly used in practice, often
accompanied by rigorous theoretical guarantees for the rate of
convergence to equilibrium. While the order of the mixing time is in
many cases well understood, e.g. for random walks on expanders, its
asymptotics remain mysterious for a wide range of problems. In
particular, in most cases it is unknown whether the convergence occurs
gradually or abruptly, the latter corresponding to the ``cutoff
phenomenon'' discovered by Diaconis and Shahshahani and independently
by Aldous in the early 80's. We will survey some aspects of this
topic, with a focus on proper colorings and independent sets in
lattices and on random walks on bounded-degree graphs.