Sharp phase transition for Gaussian percolation in all dimensions

2021/05/11 by Severo, Franco · 2 citations
#60G15 #60G60 #60K35 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2105.05219

Abstract

We consider the level-sets of continuous Gaussian fields on ℝ^d above a certain level -ℓ∈ ℝ, which defines a percolation model as ℓ varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than d (in particular, this includes the Bargmann-Fock field). Under these assumptions, we prove that the model undergoes a sharp phase transition around its critical point ℓ_c. More precisely, we show that connection probabilities decay exponentially for ℓℓ_c. This extends results recently obtained in dimension d=2 to arbitrary dimensions through completely different techniques. The result follows from a global comparison with a truncated (i.e. with finite range of dependence) and discretized (i.e. defined on the lattice εℤ^d) version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a slight change in the parameter ℓ.

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• Shadow and percolation I: discrete landscapes with independence
• Shadow and percolation II: discrete and continuous landscapes with correlations

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