Mixing time and cutoff for one dimensional particle systems

We survey recent results concerning the total-variation mixing time of the simple exclusion process on the segment (symmetric and asymmetric) and a continuum analog, the simple random walk on the simplex with an emphasis on cutoff results. A Markov chain is said to exhibit cutoff if on a certain time scale, the distance to equilibrium drops abruptly from 1 to 0. We also review a couple of techniques used to obtain these results by exposing and commenting some elements of proof. 1. A short introduction to Markov chains 1.1. Definition of a Markov chain. A stochastic process (Xt)t≥0 indexed by R+ with value in a state-space Ω is said to be a Markov process if at each time t ≥ 0, the distribution the future (Xt+u)u≥0, conditioned to that of the past (Xs)s∈[0,t] is only determined by its present state Xt. This is equivalent to say that for every bounded measurable function F : Ω+ → R there exists G : Ω → R such that