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A lower bound on the critical parameter of interlacement percolation in high dimension
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  • Published: 20 March 2010

A lower bound on the critical parameter of interlacement percolation in high dimension

  • Alain-Sol Sznitman1 

Probability Theory and Related Fields volume 150, pages 575–611 (2011)Cite this article

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Abstract

We investigate the percolative properties of the vacant set left by random interlacements on \({\mathbb{Z}^d}\), when d is large. A non-negative parameter u controls the density of random interlacements on \({\mathbb{Z}^d}\). It is known from Sznitman (Ann Math, 2010), and Sidoravicius and Sznitman (Comm Pure Appl Math 62(6):831–858, 2009), that there is a non-degenerate critical value u *, such that the vacant set at level u percolates when u < u *, and does not percolate when u > u *. Little is known about u *, however, random interlacements on \({\mathbb{Z}^d}\), for large d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, where the corresponding critical parameter can be explicitly computed, see Teixeira (Electron J Probab 14:1604–1627, 2009). We show in this article that lim inf d  u */ log d ≥ 1. This lower bound is in agreement with the above mentioned heuristics.

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Authors and Affiliations

  1. Departement Mathematik, ETH-Zentrum, 8092, Zürich, Switzerland

    Alain-Sol Sznitman

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  1. Alain-Sol Sznitman
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Correspondence to Alain-Sol Sznitman.

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Sznitman, AS. A lower bound on the critical parameter of interlacement percolation in high dimension. Probab. Theory Relat. Fields 150, 575–611 (2011). https://doi.org/10.1007/s00440-010-0284-9

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  • Received: 18 August 2009

  • Revised: 21 February 2010

  • Published: 20 March 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0284-9

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Mathematics Subject Classification (2000)

  • 60K35
  • 60G50
  • 82C41
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