= Random transposition shuffle
Choose two positions independently and uniformly from $\{1,\ldots,n\}$ and swap them. This <Markov chain> on the <symmetric group> holds with probability $1/n$ and assigns probability $2/n^2$ to each unordered <transposition>. Its <stationary distribution> is uniform, and its ordinary <spectral gap> is $2/n$ for $n\geq2$. The latter follows from the <Aldous spectral gap theorem>: the single-label <continuous-time Markov chain> has rate $2/n^2$ between every pair of positions, whose mean-zero <eigenvalues> are all $-2/n$.
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