Ratio definition of cutoff

The ratio convention for cutoff for Markov chains requires $t_{\mathrm{mix}}^{(n)}(\epsilon )/t_{\mathrm{mix}}^{(n)}(1-\epsilon )\to 1$ for every fixed $0<\epsilon <1/2$, with positive denominators eventually. Equivalently, any two fixed levels have asymptotically equal mixing times. This convention permits a bounded-time jump; definitions which additionally require diverging mixing times exclude such a jump. For nonlazy simple random walk on the complete graph $K_n$, every fixed-level mixing time is eventually one.

For finite irreducible Markov chains that are aperiodic Markov chains and reversible Markov chains, the ratio definition of cutoff and $T_n=t_{\mathrm{mix}}^{(n)}(1/4)\to \infty$ imply the Peres product condition, with ${\gamma }_n=1-{\lambda }_{2,n}$. If ${\gamma }_n{T}_n\le B$ on a subsequence, then ${\gamma }_{\ast ,n}\le {\gamma }_n$ and the eigenvalue lower bound for total variation mixing giveFor sufficiently small fixed $\epsilon$, this contradicts the limit one forced by cutoff for Markov chains. The same argument proves the stronger conclusion ${\gamma }_{\ast ,n}{T}_n\to \infty$.