Proposition 16.2 -- the $L$-reversal chain is far from mixed at $(1-\varepsilon)\frac n2\log n$

The $L$-reversal chain shuffles a circular arrangement of $n$ cards (an arrangement indexed by $\mathbb Z_n$): at each step it picks a starting position $i\in\mathbb Z_n$ and a length $k\in\{0,\dots,L-1\}$ uniformly at random and reverses the circular arc of cards from position $i$ to position $i+k$, leaving the rest fixed. Its stationary distribution is uniform. Write $P^t(x,\cdot)$ for the law after $t$ moves from the arrangement $x$, $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$ for the total variation distance, and $d_n(t)=\max_x\|P^t(x,\cdot)-\mathrm{unif}\|_{TV}$ for the chain on $n$ cards.

The theorem (Proposition 16.2 of Levin–Peres–Wilmer) asserts: for any family of maximal reversal lengths $L(n)$ with $1\le L(n)<n/2$ and any fixed $0<\varepsilon<1$, evaluating each chain at time $t_n=\bigl\lfloor(1-\varepsilon)\,\tfrac n2\log n\bigr\rfloor$ gives

Before time $\tfrac n2\log n$ the chain is asymptotically as far from uniform as possible, whatever $L$ is. The witness is again coupon-collector-flavoured: cards whose neighbourhood no reversal has yet touched retain their original neighbours in the circular order. (In the formal statement the deck sizes are enumerated as $n=m+2$, $m\to\infty$, so that $n\ge2$ automatically.)