Abstract

The vacant set of random interlacements on Z d, d ≥ 3, has non-trivial percolative properties. It is known from [18], [16], that there is a non-degenerate 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 [21]. 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 [23].

Citations