Abstract
We investigate the phase transition in a non-planar correlated percolation model with long-range dependence, obtained by considering level sets of a Gaussian free field with mass above a given height h. The dependence present in the model is a notorious impediment when trying to analyze the behavior near criticality. Alongside the critical threshold \(h_*\) for percolation, a second parameter \(h_{**} \ge h_*\) characterizes a strongly subcritical regime. We prove that the relevant crossing probabilities converge to 1 polynomially fast below \(h_{**}\), which (firmly) suggests that the phase transition is sharp. A key tool is the derivation of a suitable differential inequality for the free field that enables the use of a (conditional) influence theorem.
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Acknowledgments
The author thanks Alain-Sol Sznitman for several useful discussions and for his comments on an earlier draft of this manuscript, and both referees for their valuable remarks.
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This research was supported in part by the Grant ERC-2009-AdG 245728-RWPERCRI.
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Rodriguez, PF. A 0–1 law for the massive Gaussian free field. Probab. Theory Relat. Fields 169, 901–930 (2017). https://doi.org/10.1007/s00440-016-0743-z
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DOI: https://doi.org/10.1007/s00440-016-0743-z
Mathematics Subject Classification
- 60G15
- 60G60
- 60K35
- 82B26
- 82B27
- 82B41
- 82B43