The sign clusters of the massless Gaussian free field percolate on ${\mathbb{Z}}^d$, $d⩾3$ (and more)

We investigate the percolation phase transition for level sets of the Gaussian free field on ${\mathbb{Z}}^d$, with $d⩾3$, and prove that the corresponding critical parameter $h_{\ast }(d)$ is strictly positive for all $d⩾3$, thus settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on ${\mathbb{Z}}^d$, for all $d⩾3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of ${\mathbb{Z}}^d$ with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over ${\mathbb{Z}}^d$.