The Annals of Probability

On the critical parameter of interlacement percolation in high dimension

Alain-Sol Sznitman

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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].

Article information

Source
Ann. Probab., Volume 39, Number 1 (2011), 70-103.

Dates
First available in Project Euclid: 3 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1291388297

Digital Object Identifier
doi:10.1214/10-AOP545

Mathematical Reviews number (MathSciNet)
MR2778797

Zentralblatt MATH identifier
1210.60047

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Percolation random interlacements renormalization scheme high dimension

Citation

Sznitman, Alain-Sol. On the critical parameter of interlacement percolation in high dimension. Ann. Probab. 39 (2011), no. 1, 70--103. doi:10.1214/10-AOP545. https://projecteuclid.org/euclid.aop/1291388297


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