Mixing bounds from path coupling

Let $P$ be a Markov chain on a finite state space $V$ with stationary distribution $\pi$, and suppose the path coupling hypotheses of Theorem 14.6 hold: a connected graph $G$ on $V$ with symmetric edge lengths $\ell\ge1$, a rate $\alpha>0$, and for every edge $\{x,y\}$ of $G$ a coupling of $P(x,\cdot),P(y,\cdot)$ contracting the path metric $\rho$ (least total $\ell$-length of a connecting walk) in expectation by $e^{-\alpha}$. Write $\mathrm{diam}(V)=\max_{x,y}\rho(x,y)$ for the path-metric diameter, $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$ for the total variation distance, $d(t)=\max_x\|P^t(x,\cdot)-\pi\|_{TV}$, and $t_{\mathrm{mix}}(\varepsilon)=\min\{t:d(t)\le\varepsilon\}$.

The theorem (Corollary 14.7 of Levin–Peres–Wilmer) asserts:

1. the distance to stationarity decays geometrically: $d(t)\le e^{-\alpha t}\,\mathrm{diam}(V)$ for every $t$;
2. consequently, for every $0<\varepsilon<1$, $\;t_{\mathrm{mix}}(\varepsilon)\le\bigl\lceil(-\log\varepsilon+\log\mathrm{diam}(V))/\alpha\bigr\rceil$.

The proof is one line from path coupling: iterating the one-step contraction bounds $\rho_K(P^t(x,\cdot),\pi)$ by $e^{-\alpha t}\mathrm{diam}(V)$, and the transportation distance dominates total variation because the path metric is at least $1$ between distinct states.