Random walks on infinite percolation clusters in models with long-range correlations

Abstract

For a general class of percolation models with long-range correlations on $\mathbb{Z}^{d}$, $d\geq2$, introduced in [J. Math. Phys. 55 (2014) 083307], we establish regularity conditions of Barlow [Ann. Probab. 32 (2004) 3024–3084] that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincaré inequality. As immediate corollaries, we deduce quenched heat kernel bounds, parabolic Harnack inequality, and finiteness of the dimension of harmonic functions with at most polynomial growth. Heat kernel bounds and the quenched invariance principle of [Probab. Theory Related Fields 166 (2016) 619–657] allow to extend various other known results about Bernoulli percolation by mimicking their proofs, for instance, the local central limit theorem of [Electron. J. Probab. 14 (209) 1–27] or the result of [Ann. Probab. 43 (2015) 2332–2373] that the dimension of at most linear harmonic functions on the infinite cluster is $d+1$.

In terms of specific models, all these results are new for random interlacements at every level in any dimension $d\geq3$, as well as for the vacant set of random interlacements [Ann. of Math. (2) 171 (2010) 2039–2087; Comm. Pure Appl. Math. 62 (2009) 831–858] and the level sets of the Gaussian free field [Comm. Math. Phys. 320 (2013) 571–601] in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime for these models).