Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 215 / 3 / a / Solution

For a path $\Gamma$,by the Cauchy-Schwarz inequality. Average over ${\nu }_{xy}$, multiply by $\tilde{Q}(x,y)$, and sum. Reversing the order of summation in the definition of the two Dirichlet forms givesLet $M={\max }_x\pi (x)/\tilde{\pi }(x)$. The variational formula for variance givesTake a nonconstant eigenfunction attaining the Rayleigh quotient $\gamma$ for $P$. Then