A lower bound on the critical parameter of interlacement percolation in high dimension

We investigate the percolative properties of the vacant set left by random interlacements onZ(d), when d is large. A non-negative parameter u controls the density of random interlacements on Z(d). It is known from Sznitman (Ann Math, 2010), and Sidoravicius and Sznitman (Comm Pure Appl Math 62(6):831-858, 2009), 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(*). Little is known about u(*), however, random interlacements on Z(d), for large d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, where the corresponding critical parameter can be explicitly computed, see Teixeira (Electron J Probab 14:1604-1627, 2009). We show in this article that lim inf(d) u(*)/log d >= 1. This lower bound is in agreement with the above mentioned heuristics.