On the critical parameter of interlacement percolation in high dimension

Details On the critical parameter of interlacement percolation in high dimension On the critical parameter of interlacement percolation in high dimension

2010-03-05

arxiv.org/abs/1003.1289v2

Annals of Probability 2011, Vol. 39, No. 1, 70-103

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP545 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP545

The vacant set of random interlacements on ${\mathbb{Z}}^d$, $d\ge3$, 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 $uu_*$. 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].

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