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

In this paper, we provide general conditions on a one parameter family of random infinite subsets of Z{sup 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 Cerný and Popov [“On the internal distance in the interlacement set,” Electron. J. Probab. 17(29), 1–25 (2012)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of Antal and Pisztora [“On the chemical distance for supercritical Bernoulli percolation,” Ann Probab. 24(2), 1036–1048 (1996)] 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.
Authors:
 [1] ;  [2] ;  [3]
  1. Department of Mathematics, Columbia University, RM 614, MC 4419, 2990 Broadway, New York City, New York 10027 (United States)
  2. Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver B.C. V6T 1Z2 (Canada)
  3. Department of Mathematics, University of Leipzig, Room A409, Augustusplatz 10, 04109 Leipzig (Germany)
Publication Date:
OSTI Identifier:
22306094
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EUCLIDEAN SPACE; GRAPH THEORY; MATHEMATICAL MODELS; RANDOMNESS; SHAPE