Mixing time of a Markov chain ($t_{\mathrm{mix}}(\epsilon )$)

The $\epsilon$-mixing time is the least $t$ for which every initial state has total variation distance at most $\epsilon$ from the invariant distribution after $t$ steps.

For a finite transitive reversible chain with nontrivial eigenvalues ${\lambda }_i$,

A sequence of chains exhibits cutoff when its distance from stationarity drops from near one to near zero in a time window negligible compared with its mixing-time scale.

If adjacent states admit one-step couplings that contract a path metric by a common factor, then the induced Wasserstein distance contracts by the same factor for arbitrary starting distributions.