  We present a new framework to derandomise certain Markov chain Monte Carlo
(MCMC) algorithms. As in MCMC, we first reduce counting problems to sampling
from a sequence of marginal distributions. For the latter task, we introduce a
method called coupling towards the past that can, in logarithmic time, evaluate
one or a constant number of variables from a stationary Markov chain state.
Since there are at most logarithmic random choices, this leads to very simple
derandomisation. We provide two applications of this framework, namely
efficient deterministic approximate counting algorithms for hypergraph
independent sets and hypergraph colourings, under local lemma type conditions
matching, up to lower order factors, their state-of-the-art randomised
counterparts.
