Percolation for the Gaussian free field on the cable system: counterexamples

Abstract

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 ${\tilde{h}}_{\ast }$ is always equal to zero. On general transient graphs, two weak conditions on the graph $\mathcal{G}$ are given in arXiv:2101.05800, each of which implies one of the two inequalities ${\tilde{h}}_{\ast }\le 0$ and ${\tilde{h}}_{\ast }\ge 0.$ In this article, we give two counterexamples to show that none of these two conditions are necessary, prove that the strict inequality ${\tilde{h}}_{\ast }<0$ is typical on massive graphs with bounded weights, and provide an example of a graph on which ${\tilde{h}}_{\ast }=\infty .$ 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 $\mathbf{h}$-transform of random interlacements, and between the two-dimensional pinned free field and random interlacements.

arXiv comment41 pages

References in corpus (14)

• Phase transition and level-set percolation for the Gaussian free field
• From loop clusters and random interlacement to the free field
• Soft local times and decoupling of random interlacements
• On decoupling inequalities and percolation of excursion sets of the Gaussian free field
• On scaling limits and Brownian interlacements
• Critical exponents for a percolation model on transient graphs
• Two-dimensional random interlacements and late points for random walks
• Existence of phase transition for percolation using the Gaussian Free Field
• The first passage sets of the 2D Gaussian free field: convergence and isomorphisms
• Geometry of Gaussian free field sign clusters and random interlacements
• Cluster capacity functionals and isomorphism theorems for Gaussian free fields
• Level-set percolation for the Gaussian free field on a transient tree
• Inverting the coupling of the signed Gausssian free field with a loop soup
• Some Rigorous Results on the Phase Transition of Finitary Random Interlacement

Cited by in corpus (3)

• Critical exponents for a percolation model on transient graphs
• Cluster capacity functionals and isomorphism theorems for Gaussian free fields
• Percolation for two-dimensional excursion clouds and the discrete Gaussian free field