High-dimensional asymptotics for percolation of Gaussian free field level sets

Abstract

We consider the Gaussian free field on ${\mathbb{Z}}^d$, $d$ greater or equal to $3$, and prove that the critical density for percolation of its level sets behaves like $1/d^{1+o(1)}$ as $d$ tends to infinity. Our proof gives the principal asymptotic behavior of the corresponding critical level $h_{\ast }(d)$. Moreover, it shows that a related parameter $h_{\ast \ast }(d)\ge h_{\ast }(d)$ introduced by Rodriguez and Sznitman in arXiv:1202.5172 is in fact asymptotically equivalent to $h_{\ast }(d)$.

arXiv comment39 pages, 2 figures