Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 3 / b

For reversible chains $P,\tilde{P}$ with stationary laws $\pi ,\tilde{\pi }$, assign to every directed $\tilde{P}$-edge $(x,y)$ a path ${\Gamma }_{xy}$ of $P$-edges. Ifand $\tilde{\pi }(x)\le a\pi (x)$, the Canonical paths comparison theorem gives $\gamma (P)\ge \gamma (\tilde{P})/(aA)$, with equivalent conventions absorbing $a$ into the congestion.