Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 4 / a

For a path $\gamma =(v_0,e_1,v_1,\dots ,e_k,v_k)$, its $\ell$-length is ${\sum }_{j=1}^k\ell (e_j)$. The corresponding path metric isBecause every edge length is at least one, $\rho (x,y)\ge 1$ whenever $x\ne y$.

Let $D={\max }_{x,y}\rho (x,y)$ and let $\pi$ be the invariant distribution. Iterating the assumed Wasserstein contraction givesSince $\rho (x,y)\ge 1_{\{x\ne y\}}$, the coupling characterization of total variation distance impliesThe right side is at most $\epsilon$ whenwhich proves the claimed mixing bound.