Disorder and wetting transition: The pinned harmonic crystal in dimension three or larger

Abstract

We consider the Lattice Gaussian free field in d + 1 dimensions, d = 3 or larger, on a large box (linear size N ) with boundary conditions zero. On this field two potentials are acting: one, that models the presence of a wall, penalizes the field when it enters the lower half space and one, the pinning potential, that rewards visits to the proximity of the wall. The wall can be soft, i.e. the field has a finite penalty to enter the lower half plane, or hard when the penalty is infinite. In general the pinning potential is disordered and it gives on average a reward h ∈ R (a negative reward is a penalty): the energetic contribution when the field at site x visits the pinning region is βωx + h, {ωx} x∈Z d are IID centered and exponentially integrable random variables of unit variance and β ≥ 0. In [3] it is shown that, when β = 0 (that is, in the non disordered model), a delocalization-localization transition happens at h = 0, in particular the free energy of the system is zero for h ≤ 0 and positive for h > 0. We show that, for β = 0, the transition happens at h = hc(β) := − log E exp(βωx) and we find the precise asymptotic behavior of the logarithm of the free energy density of the system when h ց hc(β).In particular, we show that the transition is of infinite order in the sense that the free energy is smaller than any power of h − hc(β) in the neighborhood of the critical point and that disorder does not modify at all the nature of the transition. We also provide results on the behavior of the paths of the random field in the limit N → ∞.

Wetting and Layering for Solid-on-Solid I: Identification of the Wetting Point and Critical Behavior

“…For the lattice free field in higher dimension (d ≥ 3) it has been shown that there is no wetting transition (h w (β) = 0) [5]. Moreover in that case the critical behavior of the free energy and the vicinity has been identified [21].…”

Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy

“…A first step is to show convergence of the measure when having zero boundary condition to a translation invariant limit which has the right contact fraction. The proof uses essentially the same idea as those to prove a similar result for wetting of the harmonic crystal [19,Section 5].…”

Localization of a Gaussian membrane model with weak pinning potentials

“…Remark 1.5. For the ∇φ model, it has been shown in Giacomin and Lacoin (2018) that the conclusion of Theorem 1.4 yields positivity of the free energy for the case with square-well pinning together with the following hard wall condition:…”

“…Then, an argument similar to the proof of Theorem 1.1 and optimization about the mean height yield the result. We note that these types of arguments have recently been used to study the discrete Gaussian free field with pinning (see Giacomin and Lacoin, 2018). One advantage of these arguments is that we do not need to expand the pinning potential parts of (1.6) and (1.7), and therefore we do not need a random walk representation of the covariance under P N .…”