Phase Transition and Level-Set Percolation for the Gaussian Free Field

Abstract

We consider level-set percolation for the Gaussian free field on Z d , d ≥ 3, and prove that, as h varies, there is a non-trivial percolation phase transition of the excursion set above level h for all dimensions d ≥ 3. So far, it was known that the corresponding critical level h * (d) satisfies h * (d) ≥ 0 for all d ≥ 3 and that h * (3) is finite, see [2]. We prove here that h * (d) is finite for all d ≥ 3. In fact, we introduce a second critical parameter h * * ≥ h * , show that h * * (d) is finite for all d ≥ 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h > h * * . Finally, we prove that h * is strictly positive in high dimension. It remains open whether h * and h * * actually coincide and whether h * > 0 for all d ≥ 3.to a thick two-dimensional slab percolates, see above (0.12). Let us however point out that, by a result of [7] (see p. 1151 therein), the restriction of E ≥0 ϕ to Z 2 (viewed as a subset of Z d ), and, a fortiori, the restriction of E ≥h ϕ to Z 2 , when h is positive, do only contain finite connected components: excursion sets above any non-negative level do not percolate in planes. We refer to Remark 3.6 1) for more on this.We now comment on the proofs. We begin with (0.8). The key ingredient is a certain (static) renormalization scheme very similar to the one developed in Section 2 of [21] for the problem of percolation of the vacant set left by random interlacements (for a precise definition of this model, see [23], Section 1; we merely note that the two "corresponding" quantities are E ≥h ϕ and

On macroscopic holes in some supercritical strongly dependent percolation models

“…It has recently been proved that h * > 0, cf. [7], this was previously only known for large d, see [23], [10]. It is plausible, but open at the moment, that actually h = h * = h * * (possibly some progress in proving h * = h * * may come from [11]).…”

Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

“…There are three critical levels −∞ < h ≤ h * ≤ h * * < ∞ relevant to the study of the percolation of E ≥α . The strongly non-percolative regime for E ≥α corresponds to α > h * * , the strongly percolative regime corresponds to α < h, where h * * and h are defined in equation (0.6) of [20] and equation (5.3) of [22] respectively, and h * denotes the threshold for percolation of E ≥α . Moreover, it has recently been proven in [8] that h * > 0 for all dimensions d ≥ 3.…”

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.…”