A 0–1 law for the massive Gaussian free field

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 behavio...

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Bibliographic Details
Published in:Probability theory and related fields 2017-12, Vol.169 (3-4), p.901-930
Main Author: Rodriguez, Pierre-François
Format: Article
Language:English
Subjects:
Economics
Finance
Insurance
Management
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Percolation
Phase transitions
Probability
Probability Theory and Stochastic Processes
Quantitative Finance
Statistics for Business
Theoretical
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Summary: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 ∗ ∗ ≥ 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.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-016-0743-z