Quenched invariance principle for simple random walk on clusters in correlated percolation models

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We prove a quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on $${\mathbb {Z}}^d$$ , $$d\ge 2$$ , with long-range correlations introduced in (Drewitz et al. in J Math Phys 55(8):083307, 2014), solving one of the open problems from there. This gives new results for random interlacements in dimension $$d\ge 3$$ at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models.