Cluster capacity functionals and isomorphism theorems for Gaussian free fields

We investigate level sets of the Gaussian free field on continuous transient metric graphs $$\widetilde{{\mathcal {G}}}$$ G ~ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph $${\mathcal {G}}$$ G , that the capacity of sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on $$\widetilde{{\mathcal {G}}}$$ G ~ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on $$\widetilde{{\mathcal {G}}}$$ G ~ vanishes in most instances in the massless case and establishes the continuity of this phase transition in a wide range of cases, including all vertex-transitive graphs. When the sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ do not percolate, we further determine by means of isomorphism theory the exact law of the capacity of compact clusters at any height. Specifically, we derive this law from an extension of Sznitman’s refinement of Lupu’s recent isomorphism theorem relating the free field and random interlacements, proved along the way, and which holds under the sole assumption that sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ are bounded. Finally, we show that the law of the cluster capacity functionals obtained in this way actually characterizes the isomorphism theorem, i.e. the two are equivalent.