Path coupling without contraction

Path coupling is a useful technique for simplifying the analysis of a coupling of a Markov chain. Rather than defining and analysing the coupling on every pair in Ω×ΩΩ×Ω, where Ω   is the state space of the Markov chain, analysis is done on a smaller set S⊆Ω×ΩS⊆Ω×Ω. If the coefficient of contraction β   is strictly less than one, no further analysis is needed in order to show rapid mixing. However, if β=1β=1 then analysis (of the variance) is still required for all pairs in Ω×ΩΩ×Ω. In this paper we present a new approach which shows rapid mixing in the case β=1β=1 with a further condition which only needs to be checked for pairs in S  , greatly simplifying the work involved. We also present a technique applicable when β=1β=1 and our condition is not met.