  Couplings play a central role in contemporary Markov chain Monte Carlo
methods and in the analysis of their convergence to stationarity. In most
cases, a coupling must induce relatively fast meeting between chains to ensure
good performance. In this paper we fix attention on the random walk Metropolis
algorithm and examine a range of coupling design choices. We introduce proposal
and acceptance step couplings based on geometric, optimal transport, and
maximality considerations. We consider the theoretical properties of these
choices and examine their implication for the meeting time of the chains. We
conclude by extracting a few general principles and hypotheses on the design of
effective couplings.
