The approach to equilibrium is still a problem of major interest to
many scientists interested in the simulation of physical and other systems.
Most of these simulations can be described as Markov chains. In recent
years, particularly mathematicians have been intrigued by the fact that
convergence to stationarity in finite Markov chains often shows a sharp
cutoff: The variation distance between the evolving and the stationary
distribution stays close to one for a while, then drops to a small value
and, finally, vanishes exponentially. We were able to show that the cutoff
phenomen can occur in diffusive relaxation in single-well potentials where
it has a very simple explanation, and that the generic cases for the cutoff
phenomenon to occur in finite Markov chains (diffusion on the hypercube
and card shuffling) can be mapped to that particular situation. Thereby,
a simple, intuitive explanation of the phenomenon is provided.
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