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Optimal co-adapted coupling for a random walk on the hyper-complete-graph

The problem of constructing an optimal co-adapted coupling for a pair of symmetric random walks on $Z_2^d$ was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such co-adapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal co-adapted coupling for the continuous-time symmetric random walk on $K_n^d$, where $K_n$ is the complete graph with $n$ vertices. Moreover, we show that although this coupling is not maximal for any $n$ (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as $n\to\infty$.