In Markov chain Monte Carlo theory, a particular Markov chain is run for a very long time, say T, until its distribution is close enough to the stationarity. In particular, one is interested in the time T to stop the simulation and then choose the random sample. Similar problems arise in many fields including statistical physics, computer science, biology and more. It is well known that T is closely related to the mixing time of Markov chains. In recent years, for models of statistical mechanics and of theoretical computer science, there has been a flourishing of heuristic ideas to rigorously control the time.
In this talk, we will introduce the concept of cutoffs and mixing times, and discuss how they are related with each other. For illustration, the classical model, Ehrenfest processes, will be considered and latest results are described in details.