 Canonical paths comparison theorem

The Canonical paths comparison theorem transfers a spectral-gap estimate from a reference Reversible Markov chain to another chain by routing reference transitions along allowed target transitions. Let $P$ and $R$ be finite reversible transition matrices with the same positive Stationary distribution $\pi$. For every ordered pair with $R(x,y)>0$ and $x\ne y$, choose a path ${\gamma }_{xy}$ from $x$ to $y$ using positive $P$ transitions. Let $N_e(\gamma )$ count occurrences of a directed edge $e=(u,v)$ and putThen ${\mathcal{E}}_R(f,f)\le A{\mathcal{E}}_P(f,f)$ for every real-valued function $f$. Indeed, telescope $f(x)-f(y)$ along the path and apply the Cauchy-Schwarz inequality, multiply by $\pi (x)R(x,y)/2$, and sum over ordered pairs. Collecting the contribution of each directed edge gives the stated capacity-normalized congestion.

The variational Poincare inequality for a reversible Markov chain now gives ${\gamma }_P\ge {\gamma }_R/A$ for the ordinary spectral gaps. Choosing the reference chain $R(x,y)=\pi (y)$ gives ${\mathcal{E}}_R(f,f)={\mathrm{Var}}_{\pi }(f)$ and ${\gamma }_R=1$, recovering the Canonical paths Poincare bound. This makes a local-transition chain comparable to a chain that resamples its entire state in one step. The common invariant distribution and directed-edge counting convention are part of this formulation.