Non-Gaussian surface pinned by a weak potential

Abstract

Abstract. We consider a model of a two-dimensional interface of the (continuous) SOS type, with finite-range, strictly convex interactions. We prove that, under an arbitrarily weak pinning potential, the interface is localized. We consider the cases of both square well and δ potentials. Our results extend and generalize previous results for the case of nearest-neighbours Gaussian interactions in [7] and [1]. We also obtain the tail behaviour of the height distribution, which is not Gaussian.

Critical Behavior of Massless Free Field at Depinning Transition

“…Our first result complements estimates obtained in [2,8] where it was shown for d = 2 that provided V ′′ ≥ c > 0 and p( · ) satisfies (1.2), there exists a constant C > 0 (depending on p only) such that, for small enough e def = 2a…”

“…It states domination properties of the field of pinned sites by Bernoulli measures and is a substantial improvement on the results already present in [8,16]. Although the main emphasis in this paper is on the case of the (difficult) two-dimensional lattice, we include also the higher-dimensional case.…”

“…A somewhat simpler question is whether the variance stays bounded uniformly in Λ. This was shown for µ a,b Λ in the Gaussian nearest neighbor case in [9], and was finally proved in [8] much more generally, assuming only V ′′ ≥ const. > 0.…”

“…The desired inequality follows from (3.2) and, using Lemmas 5.4 and 5.5 of [8], Let Λ be a square in Z 2 , centered at the origin, and with large enough sidelength (the thermodynamic limit is taken at the end). Let B e (0) def = x ∈ Z 2 : x ∞ ≤ 1 2 e −1/2 |log e| −1/4 .…”

“…This step is a variant of the proofs given in [8,16]. Here, however, we want to keep track of the ε-dependence of the constants.…”