On Decoupling Inequalities and Percolation of Excursion Sets of the Gaussian Free Field

Abstract

We prove decoupling inequalities for the Gaussian free field on Z d , d ≥ 3. As an application, we obtain exponential decay (with logarithmic correction for d = 3) of the connectivity function of excursion sets for large values of the threshold.

Disconnection by level sets of the discrete Gaussian free field and entropic repulsion

“…Level sets of the Gaussian free field on Z d , d ≥ 3, provide an example of a percolation model with long-range dependence. Its study goes back at least to the eighties, see [3], [12], [14], and it has attracted considerable attention recently, see for instance [9], [16], [18], [21] and [6]. It is well-known that the model undergoes a phase transition at some critical level h * (d), which is both finite (see [18]) and strictly positive (as established recently in [6]) in all dimensions d ≥ 3.…”

“…For α ∈ R, we define the level set (or excursion set) above α as where B L is the closed sup-norm ball in Z d with radius L, centered at the origin, ∂B 2L is the (external) boundary of B 2L , and B L ≥α ←→ ∂B 2L stands for the event that there is a nearest-neighbor path in the excursion set E ≥α connecting B L and ∂B 2L . One can show (see [16], [18]) that h * * < ∞ for all d ≥ 3 and that for α > h * * the connectivity function P x ≥α ←→ y (with hopefully obvious notation) has a stretched exponential decay in |x − y|, the Euclidean distance between x, y ∈ Z d , which actually, is an exponential decay when d ≥ 4.…”

On macroscopic holes in some supercritical strongly dependent percolation models

“…= P G 0 ϕ≥h * * +ε ←→ ∂ int B 0, δ 4d N . (5.27)This last quantity has a stretched exponential decay in N (actually an exponential decay when d ≥ 4, with a logarithmic correction when d = 3, see Theorem 2.1 of[20]). It thus follows that of probability method, see(2.6), and the fact that for large N one has A ≥α N ⊆ { | W ≥α N | ≥ νN d } (as in (3.61)), one finds that lim infN d−2 log P G [ | W ≥α N | ≥ νN d ] ≥ − lim sup N…”

Conditional decoupling of random interlacements

“…In particular, there are decoupling inequalities similar to (1.3) and (1.5) for the Gaussian free field as well, see [12]. Notice, however, that the decoupling-with-sprinkling result for the Gaussian free field (Theorem 1.2 of [12]) is already conditional (the unconditional decoupling is obtained as a simple consequence, just by integration). On the other hand, note that the error terms in the conditional decoupling in the main result of this paper (Theorem 2.1) are much worse than that of (1.5); related to this is the fact that in the conditional setting the minimal distance between sets that permits the result to work is much bigger.…”

“…[19,20]. In particular, there are decoupling inequalities similar to (1.3) and (1.5) for the Gaussian free field as well, see [12]. Notice, however, that the decoupling-with-sprinkling result for the Gaussian free field (Theorem 1.2 of [12]) is already conditional (the unconditional decoupling is obtained as a simple consequence, just by integration).…”