  We consider a random field $\varphi:\{1,...,N\}\to \mathbb{R}$ with Laplacian
interaction of the form $\sum_iV(\Delta\varphi_i)$, where $\Delta$ is the
discrete Laplacian and the potential $V(\cdot)$ is symmetric and uniformly
strictly 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 a
defect line. It is known that this model exhibits a phase transition between a
delocalized regime $(\varepsilon<\varepsilon_c)$ and a localized one
$(\varepsilon>\varepsilon_c)$, where $0<\varepsilon_c<\infty$. In this paper we
give a precise pathwise description of the transition, extracting the full
scaling limits of the model. We show, in particular, that in the delocalized
regime 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 in
distribution toward the derivative of a symmetric stable L\'evy process of
index 2/5. Our approach is based on Markov renewal theory.
