 Pre-cutoff for Markov chains

A sequence of chains exhibits pre-cutoff if there are constants $0<A<B<\infty$ and times $t_n$ such that the worst-case distance tends to one at $A{t}_n$ and to zero at $B{t}_n$. For reversible chains, the product condition $t_{\mathrm{rel}}=o(t_{\mathrm{mix}})$ is necessary for pre-cutoff.