Scaling limit of wetting models in $1+1$ dimensions pinned to a shrinking strip

TL;DR

This paper studies the scaling limit of 1+1 dimensional wetting models on shrinking strips, showing convergence to reflected Brownian motion under specific conditions related to strip size and pinning function proximity to criticality.

Contribution

It establishes the diffusive scaling limit of wetting interfaces on shrinking strips, extending previous models to more general pinning functions and critical regimes.

Findings

Interface converges to reflected Brownian motion under diffusive scaling.

Convergence holds when strip size is o(N^{-1/2}) and pinning function is near critical.

Results apply to both general and constant pinning strip wetting models at criticality.

Abstract

We consider wetting models in $1+1$ dimensions on a shrinking strip with a general pinning function. We show that under diffusive scaling, the interface converges in law to to the reflected Brownian motion, whenever the strip size is $o(N^{-1/2})$ and the pinning function is close enough to critical value of the so-called $\delta$-pinning model of Deuschel, Giacomin, and Zambotti [DGZ05]. As a corollary, the same result holds for the constant pinning strip wetting model at criticality with $o(N^{-1/2})$ strip size.