Wetting on a wall and wetting in a well: Overview of equilibrium properties

TL;DR

This paper reviews the equilibrium properties of a wetting model with a random walk constrained above a wall, exploring phase transitions, free energy, and asymptotic behaviors in various boundary conditions and stable law domains.

Contribution

It provides new insights into the phase diagram, including the potential for a saturation transition, and derives exact asymptotics and fluctuation results for the model with general stable law domains.

Findings

Identification of a saturation phase transition alongside the wetting transition.

Explicit computation of free energy and phase diagram properties.

Asymptotic equivalence of the partition function and CLT for boundary fluctuations.

Abstract

We study the wetting model, which considers a random walk constrained to remain above a hard wall, but with additional pinning potential for each contact with the wall. This model is known to exhibit a wetting phase transition, from a localized phase (with trajectories pinned to the wall) to a delocalized phase (with unpinned trajectories). As a preamble, we take the opportunity to present an overview of the model, collecting and complementing well-known and other folklore results. Then, we investigate a version with elevated boundary conditions, which has been studied in various contexts both in the physics and the mathematics literature; it can alternatively be seen as a wetting model in a square well. We complement here existing results, focusing on the equilibrium properties of the model, for a general underlying random walk (in the domain of attraction of a stable law). First, we compute the free energy and give some properties of the phase diagram; interestingly, we find that, in addition to the wetting transition, a so-called saturation phase transition may occur. Then, in the so-called Cram\'er's region, we find an exact asymptotic equivalent of the partition function, together with a (local) central limit theorem for the fluctuations of the left-most and right-most pinned points, jointly with the number of contacts at the bottom of the well.