 Tagged-label Rayleigh quotient for adjacent transpositions

For the random adjacent transposition shuffle on $S_n$ with $n\ge 2$, the position $f(\sigma )={\sigma }^{-1}(i)$ of one label is uniform on $\{1,\dots ,n\}$ in equilibrium, so ${\mathrm{Var}}_{\pi }(f)=(n^2-1)/12$. 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 $\gamma \le 12/[n^2(n+1)]$. In particular the continuous uniform distribution on $(0,1)$ has variance $1/12$, not $1/6$.