Open Access
Translator Disclaimer
May 2017 Random walks on infinite percolation clusters in models with long-range correlations
Artem Sapozhnikov
Ann. Probab. 45(3): 1842-1898 (May 2017). DOI: 10.1214/16-AOP1103


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).


Download Citation

Artem Sapozhnikov. "Random walks on infinite percolation clusters in models with long-range correlations." Ann. Probab. 45 (3) 1842 - 1898, May 2017.


Received: 1 October 2014; Revised: 1 February 2016; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1374.60194
MathSciNet: MR3650417
Digital Object Identifier: 10.1214/16-AOP1103

Primary: 60K37
Secondary: 58J35

Keywords: Gaussian free field , Harmonic function , Harnack inequality , heat kernel , Isoperimetric inequality , local limit theorem , long-range correlations , percolation , Poincaré inequality , Random interlacements , Random walk

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 3 • May 2017
Back to Top