Towards derandomising Markov chain Monte Carlo

We present a new framework to derandomize 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 toward 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 derandomization. As an application, we provide an efficient deterministic approximate counting algorithm for hypergraph independent sets, under local lemma type conditions matching, up to lower-order factors, their state-of-the-art randomized counterparts.