On the critical parameter of interlacement percolation in high dimension

TL;DR

This paper investigates the asymptotic behavior of the critical parameter for percolation in the vacant set of random interlacements on high-dimensional integer lattices, showing it grows like the logarithm of the dimension.

Contribution

It establishes that the critical parameter $u_*$ asymptotically behaves like $ ext{log}(d)$ as the dimension $d$ increases, complementing existing bounds.

Findings

$u_*$ is asymptotically equivalent to $ ext{log}(d)$ for large $d$.

Provides an upper bound on $u_*$ that matches the known lower bound.

Shows the critical parameter's behavior parallels that on $2d$-regular trees.

Abstract

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 $u<u_*$ 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].