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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 $d$-regularexpanders 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 toinfinity. In addition, we show that this component is unique. Thissignificantly improves the result of [AC20b], where it was shown that a linearfraction of vertices is in mesoscopic components if $h<h_\star$.

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