Pre-cutoff for Markov chains
= 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 $At_n$ and to zero at $Bt_n$. For reversible chains, the product condition $t_{\mathrm{rel}}=o(t_{\mathrm{mix}})$ is necessary for pre-cutoff.