Cutoff for Markov chains: Some examples and applications

Some Markov chains converge very abruptly to their equilibrium: the total variation distance between the distribution of the chain at time t and its equilibrium measure is close to 1 until some deterministic 'cutoff time', and close to 0 shortly after. Many examples have been studied by Diaconis and his followers. Our goal is to introduce two families of examples of this phenomenon, focusing mainly on their possible applications. We present firstly samples of Markov chains for which the cutoff depends on the size of the sample. As an application, a new way of implementing Markov chain Monte-Carlo algorithms is proposed, using an explicit stopping rule based on the empirical measure of the sample. Then, we shall study Markov chains on countably many states, where the cutoff phenomenon depends on the starting point of the chain. As a particular case, a criterion of cutoff for birth and death chains on trees will be obtained. Jackson networks will show other applications of both cutoff situations.