Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 215 / 2 / b / ii / Solution

Write $r_n=t_{\mathrm{mix}}^{(n)}/t_{\mathrm{rel}}^{(n)}\to \infty$. ThenBy cutoff, at every fixed multiple $c/a_n$ the original chain is still asymptotically unmixed, while part i givesThus the new chain crosses between fixed distance levels gradually on the full scale $a_n^{-1}$: for example its $\epsilon$-mixing times are asymptotic to $a_n^{-1}\log (1/\epsilon )$. No two fixed multiples of one scale can make the limiting distance respectively one and zero, so the family has no pre-cutoff.