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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.

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