Sharp phase transition for Gaussian percolation in all dimensions

Severo, Franco

doi:10.5802/ahl.141

Severo, Franco

Sharp phase transition for Gaussian percolation in all dimensions

Annales Henri Lebesgue, Volume 5 (2022), pp. 987-1008.

Metadata

DOI10.5802/ahl.141

Keywords percolation, sharpness, phase transition, Gaussian fields

Abstract

We consider the level-sets of continuous Gaussian fields on $ℝ^{d}$ above a certain level $- ℓ \in ℝ$ , 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 $ℓ < ℓ_{c}$ and percolation occurs in sufficiently thick 2D slabs for $ℓ > ℓ_{c}$ . 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 $ε ℤ^{d}$ ) 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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