 Solution

For a finite irreducible Markov chain satisfying detailed balance, define the directed-edge capacity $Q(u,v)=\pi (u)P(u,v)$. For every ordered pair $(x,y)$ choose a path ${\eta }_{xy}$ from $x$ to $y$ using only positive-capacity edges; take the empty path for $x=y$. Let $N_e(\eta )$ count occurrences of the directed edge $e$. The canonical paths Poincare bound says thatis a valid Poincare inequality for a reversible Markov chain constant. Simple paths give $N_e\in \{0,1\}$; allowing repetitions requires the multiplicities shown.

To prove the theorem, expand the variance using two independent samples from $\pi$:Telescoping along ${\eta }_{xy}$ and applying the Cauchy-Schwarz inequality givesInterchange the finite sums, then use the definition of $\rho$:Consequently $\boxed{{\mathrm{Var}}_{\pi }(f)\le \rho \mathcal{E}(f,f),\gamma \ge 1/\rho }$. All edges here are directed, so the $1/2$ in the Dirichlet form of a Markov chain is retained consistently.