Enhanced interface repulsion from quenched hard–wall randomness

Abstract

We consider the harmonic crystal, or massless free field, ϕ = {ϕ x } x∈Z d , d ≥ 3, that is the centered Gaussian field with covariance given by the Green function of the simple random walk on Z d . Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition ϕ x to be larger than σ x , σ = {σ x } x∈Z d is an IID field (which is also independent of ϕ), for every x in a large region D N = ND ∩ Z d , with N a positive integer and D a bounded subset of R d . We are mostly motivated by results for given typical realizations of σ (quenched set-up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d + 1)-dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ 0 is heavier than Gaussian, while essentially no effect is observed if the tail is sub-Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ , leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large Deviation type asymptotics as N ∞ of the probability that ϕ lies above σ in D N .

On the Free Energy of a Gaussian Membrane Model with External Potentials

“…Additionally, by Lemma 2.1(ii) we know its precise tail asymptotics. By these facts we can proceed the similar argument to the proof of [3] which studied the problem of entropic repulsion for the i.i.d. random wall case.…”

“…Hence, the behavior of the maximum under the measure µ * ,η N closely related to the probability that the Gaussian free field stays above a random wall. This problem has been studied by [3] for the i.i.d. random wall case and by [4] for the case that the wall itself is also the Gaussian free field.…”

“…Then, for the lower bound of the maximum, we can proceed the quenched-annealed comparison and the conditioning argument which have been developed in the study of the problem of entropic repulsion (cf. [3], [4], [26], etc.). The upper bound of the maximum is given by an FKG argument combining with the precise estimate on the number of large spikes for the m η N -field.…”

“…for L large enough. We apply the quenched-annealed comparison argument used in [3] and [4]. For this purpose we prepare several notations.…”

“…. We consider the same discretization of the mean height as the proof of Proposition 2.1 of [3]. For ε > 0 small enough and j ∈ N large enough, set θ…”