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

Abstract

We consider the Gaussian free field on Zd, d≥3, and prove that the critical density for percolation of its level sets behaves like 1/d1+o(1) as d tends to infinity. Our proof gives the principal asymptotic behavior of the corresponding critical level h∗(d). Moreover, it shows that a related parameter h∗∗(d) introduced by Rodriguez and Sznitman in [23] is in fact asymptotically equivalent to h∗(d).