Sharp phase transition for Gaussian percolation in all dimensions

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 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 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.

Sharp phase transition for Cox percolation

“…Proof of the auxiliary lemmas. First, we establish Lemma 9, i.e., the domination of Inf Y z by a multiple of Inf X z , for which we adapt a coarse-graining strategy from [16]. Proof of Lemma 9.…”

“…In contrast, when applying Russo's formula for the derivative of the percolation probability, only influences with respect to random node locations appear. In order to convert one type of influence to another, we will adapt a coarse-graining strategy from [16].…”

Variance bounds for Gaussian first passage percolation

“…It boils down to sharpness of the phase transition is the supercritical regime. Very recent works have established it in general dimension only in the subcritical regime ([6] for bounded correlations, [13] with unbounded, mildly decaying correlations).…”

Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields

“…In fact, using the recent results of Severo [Sev21] in place of Theorem 1.8 below, our strategy could be adapted to prove the rigorous lower bound in d ≥ 3…”

“…), under the conditions in Theorem 1.3 and assuming g ≥ 0 (the latter condition is needed to apply the results in [Sev21]). However the matching upper bound does not follow from our strategy, essentially because one lacks a finite-size criterion for percolation in d ≥ 3 (see Section 5.1).…”