Sharp phase transition for Gaussian percolation in all dimensions

Abstract

We consider the level-sets of continuous Gaussian fields on ${\mathbb{R}}^d$ above a certain level $-\ell \in \mathbb{R}$, which defines a percolation model as $\ell$ 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 ${\ell }_c$. More precisely, we show that connection probabilities decay exponentially for $\ell <{\ell }_c$ and percolation occurs in sufficiently thick 2D slabs for $\ell >{\ell }_c$. 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 $\epsilon {\mathbb{Z}}^d$) 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 $\ell$.

19 pages