Level-set percolation of the Gaussian free field on regular graphs III: giant component on expanders

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\ge 3$. 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 $h$ has a giant component in the whole supercritical phase, that is for all $h<h_\star$, 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 [AC20b], where it was shown that a linear fraction of vertices is in mesoscopic components if $h<h_\star$.