= Detailed balance without rapid convergence
<Detailed balance> does not imply that a <Markov chain> is an <irreducible Markov chain>: the identity <transition matrix> is reversible for every <probability distribution>. Even for an <irreducible Markov chain> with <aperiodicity>, the two-state matrix with diagonal $1-\varepsilon$ and off-diagonal $\varepsilon$, $0<\varepsilon<1/2$, has uniform <stationary distribution> and nontrivial <eigenvalue> $1-2\varepsilon$. From either fixed state its <total variation distance> after $n$ steps is $\tfrac12(1-2\varepsilon)^n$, so convergence can be arbitrarily slow. Invariance, reachability and <mixing time> are distinct questions.
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