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We consider a random field $\varphi:\{1,...,N\}\to \mathbb{R}$ with Laplacianinteraction of the form $\sum_iV(\Delta\varphi_i)$, where $\Delta$ is thediscrete Laplacian and the potential $V(\cdot)$ is symmetric and uniformlystrictly convex. The pinning model is defined by giving the field a reward$\varepsilon\ge0$ each time it touches the x-axis, that plays the role of adefect line. It is known that this model exhibits a phase transition between adelocalized regime $(\varepsilon<\varepsilon_c)$ and a localized one$(\varepsilon>\varepsilon_c)$, where $0<\varepsilon_c<\infty$. In this paper wegive a precise pathwise description of the transition, extracting the fullscaling limits of the model. We show, in particular, that in the delocalizedregime the field wanders away from the defect line at a typical distance$N^{3/2}$, while in the localized regime the distance is just $O((\log N)^2)$.A subtle scenario shows up in the critical regime$(\varepsilon=\varepsilon_c)$, where the field, suitably rescaled, converges indistribution toward the derivative of a symmetric stable L\'evy process ofindex 2/5. Our approach is based on Markov renewal theory.

Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction