Title: Cutoff for Markov chains: some examples and applications.
Author: B. Ycart
Abstract:
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.
Key words: Markov chains, Cutoff, MCMC convergence, Hitting times,
Birth and death chains, Jackson networks.
AMS Subject Classification: 60 J 10, 60 J 27.