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2023 Percolation for the Gaussian free field on the cable system: counterexamples
Alexis Prévost
Author Affiliations +
Electron. J. Probab. 28: 1-43 (2023). DOI: 10.1214/23-EJP949


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.

Funding Statement

This research is supported by the Engineering and Physical Sciences Research Council (EPSRC) grant EP/R022615/1 and the Isaac Newton Trust (INT) grant G101121.


The author thanks Alexander Drewitz and Pierre-François Rodriguez for several useful discussions about the various problems solved in this article, as well as an anonymous referee for useful suggestions.


Download Citation

Alexis Prévost. "Percolation for the Gaussian free field on the cable system: counterexamples." Electron. J. Probab. 28 1 - 43, 2023.


Received: 15 December 2021; Accepted: 21 April 2023; Published: 2023
First available in Project Euclid: 2 May 2023

arXiv: 2102.07763
MathSciNet: MR4583069
Digital Object Identifier: 10.1214/23-EJP949

Primary: 60G15 , 60G60 , 60K35 , 82B43

Keywords: cable system , Gaussian free field , interlacements , isomorphism theorem , percolation

Vol.28 • 2023
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