The Cutoff Phenomenon for Ergodic Markov Processes

Abstract

We consider the cutoff phenomenon in the context of families of ergodic Markov transition functions. This includes classical examples such as families of ergodic finite Markov chains and Brownian motion on families of compact Riemannian manifolds. We give criteria for the existence of a cutoff when convergence is measured in $L^p$-norm, $1 < p < \infty$. This allows us to prove the existence of a cutoff in cases where the cutoff time is not explicitly known. In the reversible case, for $1 < p\le \infty$, we show that a necessary and sufficient condition for the existence of a max-$L^p$ cutoff is that the product of the spectral gap by the max-$L^p$ mixing time tends to infinity. This type of condition was suggested by Yuval Peres. Illustrative examples are discussed.