  We investigate the mixing properties of a model of reversible Markov chains
in random environment, which notably contains the simple random walk on the
superposition of a deterministic graph and a second graph whose vertex set has
been permuted uniformly at random. It generalizes in particular a result of
Hermon, Sly and Sousi, who proved the cutoff phenomenon at entropic time for
the simple random walk on a graph with an added uniform matching. Under mild
assumptions on the base Markov chains, we prove that with high probability the
resulting chain exhibits the cutoff phenomenon at entropic time log n/h, h
being some constant related to the entropy of the chain. We note that the
results presented here are the consequence of a work conducted for a more
general model that does not assume reversibility, which will be the object of a
companion paper. Thus, most of our proofs do not actually require
reversibility, which constitutes an important technical contribution. Finally,
our argument relies on a novel concentration result for "low-degree" functions
on the symmetric group, established specifically for our purpose but which
could be of independent interest.
