= Canonical paths Poincare bound
{title2=$\operatorname{Var}_\pi(f)\leq\rho\mathcal E(f,f)$}
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. Define
$$
\rho=\max_{e:Q(e)>0}\frac1{Q(e)}\sum_{x,y}\pi(x)\pi(y)|\eta_{xy}|N_e(\eta_{xy}).
$$
The <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 $\operatorname{Var}_\pi(f)=\frac12\sum_{x,y}\pi(x)\pi(y)(f(x)-f(y))^2$ proves the bound. This convention counts directed <edges> and retains the factor $1/2$ in the <Dirichlet form of a Markov chain>.
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