Phase Transition and Level-Set Percolation for the Gaussian Free Field

We consider level-set percolation for the Gaussian free field on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}^{d}}$$\end{document}, d ≥ 3, and prove that, as h varies, there is a non-trivial percolation phase transition of the excursion set above level h for all dimensions d ≥ 3. So far, it was known that the corresponding critical level h*(d) satisfies h*(d) ≥ 0 for all d ≥ 3 and that h*(3) is finite, see Bricmont et al. (J Stat Phys 48(5/6):1249–1268, 1987). We prove here that h*(d) is finite for all d ≥ 3. In fact, we introduce a second critical parameter h** ≥ h*, show that h**(d) is finite for all d ≥ 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h > h**. Finally, we prove that h* is strictly positive in high dimension. It remains open whether h* and h** actually coincide and whether h* > 0 for all d ≥ 3.