A 0-1 law for the massive Gaussian free field

2015/05/29 by Rodriguez, Pierre-François · 2 citations
#60G15 #60G60 #60K35 #82B26 #82B27 #82B41 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1505.08169

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_** ≥ 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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• Fractional and Integer Order Sobolev Spaces for Compact Metric Graphs
• Vector-valued Gaussian free field conditioned to avoid a ball: Entropic repulsion of the norm and Freezing of spins

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