 Canonical paths Poincare bound

Choose a directed path ${\eta }_{xy}$ of positive-probability transitions for each ordered pair of states in a finite reversible Markov chain. Write $Q(e)=\pi (u)P(u,v)$ for a directed edge $e=(u,v)$, and let $N_e(\eta )$ count its occurrences. DefineThe Cauchy-Schwarz inequality bounds $(f(x)-f(y){)}^2$ by $|{\eta }_{xy}|$ times the sum of squared differences along the path. Multiply by $\pi (x)\pi (y)/2$, sum and interchange finite sums. The identity ${\mathrm{Var}}_{\pi }(f)=\frac{1}{2}{\sum }_{x,y}\pi (x)\pi (y)(f(x)-f(y){)}^2$ proves the bound. This convention counts directed edges and retains the factor $1/2$ in the Dirichlet form of a Markov chain.