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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 $$h_*(d)$$.

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