Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

We consider the zero‐average Gaussian free field on a certain class of finite d ‐regular graphs for fixed . 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 has a giant component in the whole supercritical phase, that is for all , with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if , and together with the description of the subcritical phase from [4] establishes a fully‐fledged percolation phase transition for the model.