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On chemical distances and shape theorems in percolation models with long-range correlations. (English) Zbl 1301.82027
Summary: In this paper, we provide general conditions on a one parameter family of random infinite subsets of $${\mathbb{Z}}^d$$ to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distance. In addition, we show that these conditions also imply a shape theorem for the corresponding infinite connected component. By verifying these conditions for specific models, we obtain 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. As a byproduct, we obtain alternative proofs to the corresponding results for random interlacements in the work of J. Černý and S. Popov [Electron. J. Probab. 17, Paper No. 29, 25 p. (2012; Zbl 1245.60090)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of P. Antal and A. Pisztora [Ann. Probab. 24, No. 2, 1036–1048 (1996; Zbl 0871.60089)] for Bernoulli percolation. Finally, as a corollary, we derive new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.{
©2014 American Institute of Physics}

##### MSC:
 82B43 Percolation 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
##### Keywords:
percolation models; random walk
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##### References:
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