Cluster capacity functionals and isomorphism theorems for Gaussian free fields

Abstract

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.

Critical exponents for a percolation model on transient graphs

“…where d(•, •) refers to the distance introduced below (1.2). The condition (G ν ) actually implies (1.8), as follows by combining Corollary 3.3,1) and Lemma 3.4,2) in [12].…”

“…As shown in our companion article [12], see Theorem 1.1,1) and Lemma 3.4,2) therein (see also Remark 3.2,1) below), the condition (1.8) is generic in that it is satisfied for a wide range of graphs G, including e.g. all vertex-transitive graphs (with unit weights).…”

“…This diffusion can be defined in terms of its Dirichlet form or directly constructed from a corresponding discrete-time Markov chain by adding independent Brownian excursions on the edges beginning at a vertex; we refer e.g. to Section 2.1 of [12] for details regarding the construction of the measure P x . We denote by g(•, •) the Green function associated to this Brownian motion, that is the density of the local times of X • at infinity with respect to the natural Lebesgue measure on G, which attaches length 1/2λ x,y to every cable.…”

“…Our replacements for (i) and (ii) harvest a powerful and profound link between ϕ and the random interlacement sets (I u ) u>0 on G, see e.g. Section 2.5 in [12] for their precise definition in the present context. The random sets I u ⊂ G, u > 0 can be jointly defined in such a way that I u is increasing in u and its law is characterized by the property that (1.29)…”

“…Theorem 1.1, see in particular (1.10), implies that in very broad generality -namely assuming (1.8) only, which guarantees that a = 0(= a * ) is critical, cf. [12] for a thorough investigation into the validity of this assumption, and establishes the phase transition as second order -, one has…”