 Solution

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