On level-set percolation for the Gaussian free field in high dimensions

We investigate the percolation model obtained by considering level sets of the Gaussian free field on the d-dimensional lattice above a given height h. It has recently been proved that, as h varies, this model exhibits a non-trivial percolation phase transition in all dimensions d greater or equal to 3. We show that the associated critical density behaves like $1/d^{1+o(1)}$ as d goes to infinity. The proof gives the (principal) asymptotics of the corresponding critical height $h_{\ast }(d)$. Moreover, it shows that a related parameter $h_{\ast \ast }(d)$, which characterizes a strongly subcritical regime, is in fact asymptotically equivalent to $h_{\ast }(d)$.