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

Abstract

We provide a complete description of the low temperature wetting transition for the two dimensional Solid-On-Solid model. More precisely we study the integer-valued field (φ(x)) x∈Z 2 , associated associated to the energy functionalIt is known since the pioneering work Chalker [14] of that for every β, there exists hw(β) > 0 delimiting a transition between a delocalized phase (h < hw(β)) where the proportion of points at level zero vanishes, and a localized phase (h > hw(β)) where this proportion is positive. We prove in the present paper that for β sufficiently large we have hw(β) = log e 4β e 4β − 1 .Furthermore we provide a sharp asymptotic for the free energy at the vicinity of the critical point: we show that close to hw(β), the free energy is approximately piecewise affine and that the points of discontinuity for the derivative of the affine approximation forms a geometric sequence accumulating on the right of hw(β). This asymptotic behavior provides a strong evidence for the conjectured existence of countably many "layering transitions" at the vicinity of the critical point, corresponding to jumps for the typical height of the field.

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

“…The existence of Gibbs states when the free energy is also proved, together with uniqueness on intervals where the free energy is differentiable, and non-uniqueness at points of non differentiability. Combined with the results obtained in [26] concerning the value critical point h w (β) and the sharp asymptotics for the free energy, this yields a complete picture of the systems behavior.…”

“…where the limit can be taken over any sequence of finite sets (Λ N ) N ≥0 such such ratio between the cardinality of Λ N and that of its boundary vanishes. A justification of the existence of the limit is given in the introduction of [26]. We used | · | to denote the cardinality of a set, and we keep this notation in the remainder of the paper.…”

“…The asymptotic behavior for the free energy. In previous work [26], we established the value of h w (β) answering a question left open since the pioneering work of Chalker [14], and we were able to describe the asymptotic behavior of f(β, h) close to the critical point. To state this result we need to introduce a few quantities.…”

“…For this reason, a particular attention has been given to results obtained for SOS concerning the transition from to rigid interfaces at low temperature to rough ones at high temperature [9,18,27], and to the study of layering and wetting transitions in presence of an interaction with a wall (in a wetting or pre-wetting setup) [4,5,7,11,12,14,15] . We refer to the recent review [24] for a richer introduction to effective interface models as well to the introduction of [26] for additional motivation and references.…”

Confinement of Brownian polymers under geometric area tilts

“…Indeed, the main motivation comes from the study of the fluctuations of level lines in the random surface separating lowtemperature phases in the solid-on-solid (SOS) approximation of the 3D Ising model. Before describing our model and main results, let us give more details about the context where the model naturally arises, we refer to [4,5,15,19], for further information.…”