On chemical distances and shape theorems in percolation models with long-range correlations

TL;DR

This paper establishes conditions under which infinite clusters in long-range correlated percolation models have chemical distances comparable to Euclidean distances, proving a shape theorem and applying results to various models including random interlacements and Gaussian free fields.

Contribution

It provides general conditions for long-range correlated percolation models to exhibit shape theorems and comparable chemical and Euclidean distances, extending previous Bernoulli percolation results.

Findings

Proved shape theorem for chemical distance balls.

Established conditions for unique infinite cluster with Euclidean comparable distances.

Applied results to random interlacements and Gaussian free fields.

Abstract

In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. We also obtain alternative proofs to the main results in arXiv:1111.3979. Finally, as a corollary, we obtain new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.