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

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