  We prove a {\it{quenched}} large deviation principle (LDP) for a simple
random walk on a supercritical percolation cluster (SRWPC) on $\mathbb Z^d$
($d\geq 2$). The models under interest include classical Bernoulli bond and
site percolation as well as models that exhibit long range correlations, like
the random cluster model, the random interlacement and {the vacant set of
random interlacements} (for $d\geq 3$) and the level sets of the Gaussian free
field ($d\geq 3$).
  Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan
([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift,
and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding
elliptic random walks in random environment, we take the point of view of the
moving particle and prove a large deviation principle for the quenched
distribution of the {\it{pair empirical measures}} of the environment Markov
chain in the non-elliptic case of SRWPC . Via a contraction principle, this
reduces easily to a quenched LDP for the distribution of the mean velocity of
the random walk and both rate functions admit explicit variational formulas.
  The main difficulty in our set up lies in the inherent non-ellipticity as
well as the lack of {\it{translation-invariance}} stemming from conditioning on
the fact that the origin belongs to the infinite cluster. We develop a unifying
approach for proving quenched large deviations for SRWPC based on exploiting
coercivity properties of the relative entropies in the context of convex
variational analysis, combined with input from ergodic theory and invoking
geometric properties of the supercritical percolation cluster.
