Separation vs total variation: $s(2t)\le 1-(1-\bar d(t))^2$

Let $P$ be a Markov chain on a finite state space $V$ with stationary distribution $\pi$, reversible with respect to it (detailed balance: $\pi(x)P(x,y)=\pi(y)P(y,x)$). Two ways of measuring distance from stationarity at time $t$: the maximal separation distance

which is small only when every transition probability has caught up with its stationary value, and the worst pairwise total variation distance $\bar d(t)=\max_{x,y}\|P^t(x,\cdot)-P^t(y,\cdot)\|_{TV}$, with $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$ (Mission II).

The theorem (Lemma 19.3, Aldous–Diaconis; Levin–Peres–Wilmer) asserts: for every time $t$,

Separation at twice the time is controlled by total variation at the original time: once the chain is well mixed in total variation, running it for the same time again brings every individual transition probability up to nearly its stationary value. The proof writes $P^{2t}(x,y)$ as a sum over midpoints, applies reversibility to fold the two halves, and uses Cauchy–Schwarz. In this mission the lemma is the bridge from cover-time estimates (which control $\bar d$ for the lamplighter chain) to the separation bounds in the lamplighter mixing theorem.