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January 2011 On the critical parameter of interlacement percolation in high dimension
Alain-Sol Sznitman
Ann. Probab. 39(1): 70-103 (January 2011). DOI: 10.1214/10-AOP545

Abstract

The vacant set of random interlacements on ℤd, d≥3, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039–2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831–858] that there is a nondegenerate critical value u such that the vacant set at level u percolates when u<u and does not percolate when u>u. We derive here an asymptotic upper bound on u, as d goes to infinity, which complements the lower bound from Sznitman [Probab. Theory Related Fields, to appear]. Our main result shows that u is equivalent to log d for large d and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d-regular trees, which has been explicitly computed in Teixeira [Electron. J. Probab. 14 (2009) 1604–1627].

Citation

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Alain-Sol Sznitman. "On the critical parameter of interlacement percolation in high dimension." Ann. Probab. 39 (1) 70 - 103, January 2011. https://doi.org/10.1214/10-AOP545

Information

Published: January 2011
First available in Project Euclid: 3 December 2010

zbMATH: 1210.60047
MathSciNet: MR2778797
Digital Object Identifier: 10.1214/10-AOP545

Subjects:
Primary: 60G50 , 60K35 , 82C41

Keywords: high dimension , percolation , Random interlacements , renormalization scheme

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 1 • January 2011
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