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High-dimensional asymptotics for percolation of Gaussian free field level sets. (English) Zbl 1321.60207
Summary: We consider the Gaussian free field on \(\mathbb{Z}^d\), \( d\geq3\), 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 \(h_*(d)\). Moreover, it shows that a related parameter \(h_{**}(d)\) introduced by P.-F. Rodriguez and A.-S. Sznitman [Commun. Math. Phys. 320, No. 2, 571–601 (2013; Zbl 1269.82028)] is in fact asymptotically equivalent to \(h_*(d)\).

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G60 Random fields
60G15 Gaussian processes
82B43 Percolation
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