Phase transition for level-set percolation of the membrane model in dimensions $d \geq 5$

Abstract

We consider level-set percolation for the Gaussian membrane model on Z d , with d ≥ 5, and establish that as h ∈ R varies, a non-trivial percolation phase transition for the level-set above level h occurs at some finite critical level h * , which we show to be positive in high dimensions. Along h * , two further natural critical levels h * * and h are introduced, and we establish that −∞ < h ≤ h * ≤ h * * < ∞, in all dimensions. For h > h * * , we find that the connectivity function of the level-set above h admits stretched exponential decay, whereas for h < h, chemical distances in the (unique) infinite cluster of the level-set are shown to be comparable to the Euclidean distance, by verifying conditions identified by Drewitz, Ráth and Sapozhnikov [14] for general correlated percolation models. As a pivotal tool to study its level-set, we prove novel decoupling inequalities for the membrane model. * (d) ∈ (0, ∞), * (d), the unique infinite cluster is "well-behaved" in the sense that its chemical distances are close to the Euclidean distance, and the simple random walk on it fulfills a quenched invariance principle, with Gaussian heat kernel bounds. We refer to [17] for an even more precise investigation of connection probabilities in the entire off-critical regime h = h GFF * (d) for the level-set percolation of the GFF.The membrane model on Z d , d ≥ 5, may be seen as a variant of the GFF, in which the gradient structure in the corresponding Gibbs measure is replaced by the discrete Laplacian. This choice gives rise to a discrete interface that favors constant curvature, and models of this type are used in the physics literature to characterize thermal fluctuations of biomembranes formed by lipid bilayers (see, e.g. [23,25]). While the membrane model retains some crucial properties of the GFF − in particular one still has a domain Markov property − it lacks some key features which have made the mathematical investigation of the GFF tractable, such as an elementary finite-volume random walk representation or a finite-volume FKG inequality. A number of classical results for the GFF have been verified in the context of the membrane model, in particular the behavior of its maximum and entropic repulsion by a hard wall (see [6,8,11,19,20,21,34,36]), whereas questions concerning its level-set percolation have remained open. In the present article, we make progress in this direction by establishing that a phase transition occurs at a finite level h * (d) in d ≥ 5, and by characterizing parts of its subcritical and supercritical regimes, similar in spirit to the above mentioned program for the GFF. A key tool in our proofs is a decoupling inequality for the membrane model, which we derive in Section 3 akin to what was done for the GFF in [29]. This decoupling inequality is instrumental to prove the finiteness of two further critical parameters h(d)(≤ h * (d) ≤)h * * (d), characterizing a strongly percolative regime h < h(d) and a strongly non-percolative regime h > h * * (d), respectively.

The intermediate level-sets of the four-dimensional membrane model

“…There is also an on-going work [12] by the first author and collaborators investigating the extremal process in four dimension, aiming at establishing similar characterizations as those for the planar GFF. In supercritical dimensions, [19] studies the percolation of the level-sets, inspired by [6] for similar results on supercritical GFF. The membrane model also has a scaling limit, which is introduced and studied in [8] by Cipriani, Dan and Hazra.…”

“…Property (4) is trivial by decomposing h DN ∪ DN by the independent sum of h DN and h DN . Property(6) is trivially true by translation invariance. Now we turn to property(5).…”