This enables us, in Theorems \[th:glaubercompare\] and \[th:glaubergrid\], to prove what is needed for Theorem \[th:main2\] part (i), and Theorem \[th:main3\] parts (i) and (ii). Note that the final proofs of Theorems \[th:main2\] and \[th:main3\] are left until we have all the pieces, at the end of Section \[se:tormix\].

Coupling {#se:coupling}
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The notion of coupling (more specifically path coupling [@BubleyDyer]) lies at the heart of our proofs of upper bounds for mixing times. We give the basic setup in this section.

Let $\mathcal{M} = (X_t)$ be a Markov chain with transition matrix $P$. A *coupling* for $\mathcal{M}$ is a stochastic process $(A_t,B_t)$ on $\Omega \times \Omega$ such that each of $(A_t)$ and $(B_t)$, considered independently, is a faithful copy of $(X_t)$. Since all our processes are time-homogeneous, a coupling is determined by its transition matrix: given elements $(a,b)$ and $(a',b')$ of $\Omega \times \Omega$, let $P'((a,b),(a',b'))$ be the probability that $(A_{t+1},B_{t+1}) = (a',b')$ given that $(A_t,B_t) = (a,b)$. Since $(A_t,B_t)$ is a coupling, for each fixed $(a,b) \in \Omega \times \Omega$, we have $$\begin{aligned}
\sum_{b' \in \Omega}P'((a,b),(a',b')) &= P(a,a') \quad \text{ for all } a' \in \Omega; \\
\sum_{a' \in \Omega}P'((a,b),(a',b')) &= P(b,b') \quad \text{ for all } b' 