  In this paper we provide general conditions on a one parameter family of
random infinite subsets of Z^d to contain a unique infinite connected component
for which the chemical distances are comparable to the Euclidean distances,
focusing primarily on models with long-range correlations. Our results are in
the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We
also prove a shape theorem for balls in the chemical distance under such
conditions. Our general statements give novel results about the structure of
the infinite connected component of the vacant set of random interlacements and
the level sets of the Gaussian free field. We also obtain alternative proofs to
the main results in arXiv:1111.3979. Finally, as a corollary, we obtain new
results about the (chemical) diameter of the largest connected component in the
complement of the trace of the random walk on the torus.
