Theorem 5.2 and Corollary 5.3 -- the coupling bound

Let $P$ be a Markov chain on a finite state space $V$ with stationary distribution $\pi$, and write $P^t(x,\cdot)$ for the distribution at time $t$ started at $x$, $\|\mu-\nu\|_{TV}=\max_{A\subseteq V}|\mu(A)-\nu(A)|$ for the total variation distance, and $d(t)=\max_x\|P^t(x,\cdot)-\pi\|_{TV}$ for the worst-case distance to stationarity. A Markovian coupling of $P$ is a Markov chain on ordered pairs of states each of whose two coordinates, viewed on its own, moves according to $P$, and which keeps the two coordinates together once they coincide. For such a coupling started at the pair $(x,y)$, the coupling time $\tau_{\mathrm{couple}}$ is the first time the pair chain reaches the diagonal $\{(v,v)\}$ — the moment the two copies meet.

Suppose a Markovian coupling $Q_{x,y}$ of $P$ is given for every pair of starting states. The theorem (Theorem 5.2 and Corollary 5.3 of Levin–Peres–Wilmer) asserts:

1. for every pair $x,y$ and time $t$, the rows of $P^t$ are close whenever the coupling has probably met: $\bigl\|P^t(x,\cdot)-P^t(y,\cdot)\bigr\|_{TV}\le\mathbb P_{x,y}\{\tau_{\mathrm{couple}}>t\}$;
2. consequently $d(t)\le\max_{x,y}\mathbb P_{x,y}\{\tau_{\mathrm{couple}}>t\}$.

This is the engine of the coupling method: to bound mixing, build a coupling that meets fast.