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

Sznitman, Alain-Sol

In: Probability Theory and Related Fields, 2011, vol. 150, no. 3-4, p. 575-611

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    Title
    • A lower bound on the critical parameter of interlacement percolation in high dimension
    Author
    • Sznitman, Alain-Sol. Departement Mathematik, ETH-Zentrum, 8092, Zürich, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Probability Theory and Related Fields, 2011, vol. 150, no. 3-4, p. 575-611. Springer-Verlag
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:319046
    Summary
    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 infdu */ log d ≥ 1. This lower bound is in agreement with the above mentioned heuristics