= Canonical paths comparison theorem
{c}
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 put
$$
A=\max_{e:\,P(u,v)>0}\frac{1}{\pi(u)P(u,v)}
\sum_{x,y}\pi(x)R(x,y)|\gamma_{xy}|N_e(\gamma_{xy}).
$$
Then $\mathcal E_R(f,f)\leq A\mathcal E_P(f,f)$ 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 $\gamma_P\geq\gamma_R/A$ for the ordinary <spectral gaps>. Choosing the reference chain $R(x,y)=\pi(y)$ gives $\mathcal E_R(f,f)=\operatorname{Var}_\pi(f)$ 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.
Back to article page
