Cracking the Cutoff Window

C. Lancia
11 June 2012
The cutoff phenomenon is the abrupt convergence to stationarity of a Markov chain. It is characterized by a narrow window centered around a cutoff-time in which the distance from stationarity suddenly drops from 1 to 0.

All the examples in which cutoff was detected clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon. In the case of birth- and- death chains this mechanism is fairly well understood.

I will present a possible generalization of this picture to more general systems and show that there are two sources of randomness contributing to the size of the cutoff window. One is related to the drift towards the relevant quantiles of $\pi$ and the other to the thermalization in that region of the state space.

For one-dimensional systems a sufficiently strong drift ensures that the thermalization is under control but for higher-dimensional models the thermalization contribution can grow wide the cutoff window and even destroy completely the phenomenon.
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