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.
For massless vertex-transitive transient graphs, the percolation phase transition for the level sets of the Gaussian free field on the associated continuous cable system is particularly well understood, and in particular the associated critical parameter h * is always equal to zero. On general transient graphs, two weak conditions on the graph G are given in [12], each of which implies one of the two inequalities h * ≤ 0 and h * ≥ 0. In this article, we give two counterexamples to show that none of these two conditions are necessary, prove that the strict inequality h * < 0 is typical on massive graphs with bounded weights, and provide an example of a graph on which h * = ∞. On the way, we obtain another characterization of random interlacements on massive graphs, as well as an isomorphism between the Gaussian free field and the Doob h-transform of random interlacements, and between the two-dimensional pinned free field and random interlacements.
Contents
We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the threedimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.
We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.
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