Cutoff is a step-function profile

Consider a family of chains: for each $n$, an irreducible and aperiodic chain $P^{(n)}$ on a finite state space with stationary distribution $\pi_n$ — irreducible and aperiodic so that it does converge to $\pi_n$ and its mixing times are genuine finite times. Write $d_n(t)=\max_x\|P^{(n)t}(x,\cdot)-\pi_n\|_{TV}$ for the worst-case total variation distance to stationarity ($\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$), $t^{(n)}_{\mathrm{mix}}(\varepsilon)=\min\{t:d_n(t)\le\varepsilon\}$, and $t^{(n)}_{\mathrm{mix}}=t^{(n)}_{\mathrm{mix}}(1/4)$. The family has a cutoff when $t^{(n)}_{\mathrm{mix}}(\varepsilon)/t^{(n)}_{\mathrm{mix}}(1-\varepsilon)\to1$ for every $0<\varepsilon<1$: the times to mix well and to mix barely agree to leading order.

The theorem (Lemma 18.1 of Levin–Peres–Wilmer) asserts that cutoff is equivalent to the distance profile converging to a step function on the $t_{\mathrm{mix}}$ time scale: the family has a cutoff if and only if for every $c>0$

1. $c<1$ implies $d_n\bigl(\lfloor c\,t^{(n)}_{\mathrm{mix}}\rfloor\bigr)\to1$ — just before the mixing time the family is asymptotically unmixed;
2. $c>1$ implies $d_n\bigl(\lfloor c\,t^{(n)}_{\mathrm{mix}}\rfloor\bigr)\to0$ — just after it, asymptotically mixed.

This equivalence is the working definition of cutoff in practice: sharp upper and lower bounds on $d_n$ at times $c\,t_{\mathrm{mix}}$ (as produced for the hypercube in this mission) are exactly what the right-hand side asks for.