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

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.

33 pages, 2 figures

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• Random walks on supercritical percolation clusters
• Soft local times and decoupling of random interlacements
• On the uniqueness of the infinite cluster of the vacant set of random interlacements
• On decoupling inequalities and percolation of excursion sets of the Gaussian free field
• On the easiest way to connect $k$ points in the Random Interlacements process
• Large deviations for simple random walk on percolations with long-range correlations

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• Geometry of Gaussian free field sign clusters and random interlacements
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• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Quenched large deviations for simple random walks on percolation models including long-range correlations
• Analyticity of Gaussian free field percolation observables
• Distances in $\frac{1}{|x-y{|}^{2d}}$ percolation models for all dimensions
• On (non-)monotonicity and phase diagram of finitary random interlacement
• Phase transition for level-set percolation of the membrane model in dimensions $d\ge 5$
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• First passage percolation, local uniqueness for interlacements and capacity of random walk
• All spatial random graphs with weak long-range effects have chemical distance comparable to Euclidean distance
• Behavior of the distance exponent for $\frac{1}{|x-y{|}^{2d}}$ long-range percolation
• Absence of weak disorder for directed polymers on supercritical percolation clusters
• Chemical distance in geometric random graphs with long edges and scale-free degree distribution