Mirror coupling under an involution
= Mirror coupling under an involution
Suppose an <involution> $R$ of a <Markov chain> satisfies $P(Rx,Ry)=P(x,y)$. Start one chain at $x$, set the other equal to its image under $R$ until the first <hitting time> of the fixed-point set of $R$, and subsequently use identical transitions. This is a <coalescing coupling>, because the symmetry preserves each marginal <transition matrix> and both chains coincide at that <hitting time>.