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

For a reversible transition matrix $P$ on $L^2(\pi )$, the spectral gap isEquivalently for a lazy chain,

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.

Let $B=2{\mathbb{Z}}_{2n}^d$. The chain induced on $B$ by the lazy walk on $A$ has uniform stationary distribution, and an excursion from one even vertex can return there or reach only one of its $2d$ even lattice neighbors at displacement $\pm 2e_i$. Its transition conductances are bounded above by constants depending only on $d$. Testing its Dirichlet quotient withtherefore givesIndeed neighboring values differ by $O(n^{-1})$, while the variance of $f$ is bounded below uniformly. The supplied trace-chain theorem gives ${\gamma }_B\ge {\gamma }^{(2n)}(A)$, and hence