Geometry of Gaussian free field sign clusters and random interlacements

Abstract

For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters of the Gaussian free field on $G$ 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 belonging to this class include regular lattices like ${\mathbb{Z}}^d$, for $d⩾3$, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman in arXiv:0704.2560, and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from arXiv:1010.1490.

arXiv comment79 pages, 2 figures

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Cited by in corpus (11)

• Equality of critical parameters for percolation of Gaussian free field level-sets
• Critical exponents for a percolation model on transient graphs
• On the radius of Gaussian free field excursion clusters
• Cluster capacity functionals and isomorphism theorems for Gaussian free fields
• On bulk deviations for the local behavior of random interlacements
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Percolation for the Gaussian free field on the cable system: counterexamples
• Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets
• Generating Galton-Watson trees using random walks and percolation for the Gaussian free field
• First passage percolation, local uniqueness for interlacements and capacity of random walk
• Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension $d\ge 3$