= Tagged-label Rayleigh quotient for adjacent transpositions
{title2=$\gamma\leq12/[n^2(n+1)]$}
For the <random adjacent transposition shuffle> on $S_n$ with $n\geq2$, the position $f(\sigma)=\sigma^{-1}(i)$ of one label is uniform on $\{1,\ldots,n\}$ in equilibrium, so $\operatorname{Var}_\pi(f)=(n^2-1)/12$. Each adjacent <transposition> moves the label with probability $2/n$, giving $\mathcal E(f,f)=(n-1)/n^2$. The <Poincare inequality for a reversible Markov chain> variational formula therefore gives $\gamma\leq12/[n^2(n+1)]$. In particular the continuous <uniform distribution> on $(0,1)$ has <variance> $1/12$, not $1/6$.
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