The scaling limits of the non critical strip wetting model

TL;DR

This paper investigates the phase transition in the strip wetting model, a one-dimensional random walk with rewards on a strip, providing a detailed description of the transition and its scaling limits using Markov renewal theory.

Contribution

It offers a precise pathwise description of the phase transition and determines the full scaling limits of the model, advancing understanding of wetting phenomena.

Findings

Identification of a phase transition between localized and delocalized regimes.

Explicit characterization of the critical point _{c}^{a} depending on parameters.

Full scaling limits of the model are derived.

Abstract

The strip wetting model is defined by giving a (continuous space) one dimensionnal random walk $S$ a reward $\gb$ each time it hits the strip $\R^{+} \times [0,a]$ (where $a$ is a positive parameter), which plays the role of a defect line. We show that this model exhibits a phase transition between a delocalized regime ($\gb < \gb_{c}^{a}$) and a localized one ($\gb > \gb_{c}^{a}$), where the critical point $\gb_{c}^{a} > 0$ depends on $S$ and on $a$. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. Our approach is based on Markov renewal theory.