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

We consider level-set percolation for the Gaussian free field on , d a parts per thousand yen 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 a parts per thousand yen 3. So far, it was known that the corresponding critical level h (*)(d) satisfies h (*)(d) a parts per thousand yen 0 for all d a parts per thousand yen 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 a parts per thousand yen 3. In fact, we introduce a second critical parameter h (**) a parts per thousand yen h (*), show that h (**)(d) is finite for all d a parts per thousand yen 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.