Level-set percolation of the Gaussian free field on regular graphs II: finite expanders

We consider the zero-average Gaussian free field on a certain class of finite d-regular graphs for fixed d >= 3. This class includes d-regular expanders of large girth and typical realisations of random d-regular graphs. We show that the level set of the zero-average Gaussian free field above level h(*), exhibits a phase transition at level which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite d-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level h does not contain any connected component of larger than logarithmic size whenever h > h(*), and on the contrary, whenever h < h(*), linear fraction of the vertices is contained in connected components of the level set above level h having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase h < h(*), as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level h. The proofs in this article make use of results from the accompanying paper [2].