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Journal Articles Stochastic Processes and their Applications Year : 2024

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

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 Cramér’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.
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Dates and versions

hal-04210456 , version 1 (18-09-2023)

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Quentin Berger, Brune Massoulié. Wetting on a wall and wetting in a well: Overview of equilibrium properties. Stochastic Processes and their Applications, inPress, 170, pp.104299. ⟨10.1016/j.spa.2024.104299⟩. ⟨hal-04210456⟩
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