Birth-and-death Markov chains exhibit a sharp cutoff in their convergence
to equilibrium if suitable drift conditions are imposed on the transition
rates. The cutoff behavior appears to be closely related to the fact that
the stationary distribution is mostly concentratedon a region A whose
diameter is much smaller than the size of the state space.
Then the cutoff-time is understood to be the effective amount of time
necessary to reach A. The aim of this work is to extend this picture to
the apparently unlike case of Markov chains with highly symmetric state
space, for which the equilibrium measure is uniform.
As a matter of fact, if it is possible to project the state space onto
equivalence classes such that the entropy of the system is highly
concentrated on a few of them, the behavior of the lumped chain will be
analogous to the one of a birth and death process with the role of
stationary distribution played by the entropy.
I will review some applications of this result.