# Isomorphism theorems and the sign cluster geometry of the Gaussian free field

We consider the Gaussian free field (GFF) on a large class of transient weighted graphs G, and prove that its sign clusters contain an infinite connected component. In fact, we show that the sign clusters fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs G belonging to this class include cases in which the random walk on G exhibits anomalous diffusive behavior. Among other things, our proof exploits a certain relation (isomorphism theorem) relating the GFF to random interlacements, which form a Poissonian soup of bi-infinite random walk trajectories.
Our findings also imply the existence of a nontrivial percolating regime for the vacant set of random interlacements on G.

Based on joint work with A. Prévost and A. Drewitz.