On the critical parameter of interlacement percolation in high dimension

Abstract

The vacant set of random interlacements on ${\mathbb{Z}}^d$, $d\ge 3$, 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 $u_{\ast }$ such that the vacant set at level $u$ percolates when $u<u_{\ast }$ and does not percolate when $u>u_{\ast }$. We derive here an asymptotic upper bound on $u_{\ast }$, as $d$ goes to infinity, which complements the lower bound from Sznitman [Probab. Theory Related Fields, to appear]. Our main result shows that $u_{\ast }$ 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].

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

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• Percolation on finite graphs and isoperimetric inequalities
• On the uniqueness of the infinite cluster of the vacant set of random interlacements
• Upper bound on the disconnection time of discrete cylinders and random interlacements