On the critical parameter of interlacement percolation in high dimension

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].