  We show quenched large deviations for the simple random walk on a certain
class of percolations with long-range correlations. This class contains the
supercritical Bernoulli percolations, the model considered by Drewitz, R'ath
and Sapozhnikov and the random-cluster model up to the slab critical point. Our
result is an extension of Kubota's result for the supercritical Bernoulli
percolations. We also state a shape theorem for the chemical distance, which is
an extension of Garet and Marchand's result for the supercritical Bernoulli
percolations.
