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

2013/10/31 by Alexander Drewitz, Pierre-François Rodriguez
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP #msc:60G15 #msc:60G60 #msc:60K35 #msc:82B43

paper · pdf

published as Electron. J. Probab. 20(47), 1-39 (2015) · 39 pages, 2 figures

arxiv created 2015/04/27 · arxiv updated 2015/04/28

Abstract

We consider the Gaussian free field on 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_*(d). Moreover, it shows that a related parameter h_**(d) ≥ h_*(d) introduced by Rodriguez and Sznitman in arXiv:1202.5172 is in fact asymptotically equivalent to h_*(d).

Citations

• Phase transition and level-set percolation for the Gaussian free field