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

TL;DR

This paper establishes regularity and geometric properties of the infinite percolation cluster in models with long-range correlations, leading to various heat kernel and harmonic function results, extending known percolation theory to new models.

Contribution

It proves regularity conditions for mesoscopic balls in infinite clusters of correlated percolation models, enabling extension of key probabilistic results to these models.

Findings

Regular volume growth of large balls in the cluster

Weak Poincaré inequality holds on mesoscopic scales

Finiteness of harmonic functions with polynomial growth

Abstract

For a general class of percolation models with long-range correlations on $\mathbb Z^d$, $d\geq 2$, introduced in arXiv:1212.2885, we establish regularity conditions of Barlow arXiv:math/0302004 that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincar\'e 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 arXiv:1310.4764 allow to extend various other known results about Bernoulli percolation by mimicking their proofs, for instance, the local central limit theorem of arXiv:0810.2467 or the result of arXiv:1111.4853 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 $d\geq 3$, as well as for the vacant set of random interlacements arXiv:0704.2560, arXiv:0808.3344 and the level sets of the Gaussian free field arXiv:1202.5172 in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime for these models).