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We consider a random field $\varphi:\{1,...,N\}\to\mathbb{R}$ as a model fora linear chain attracted to the defect line $\varphi=0$, that is, the x-axis.The free law of the field is specified by the density$\exp(-\sum_iV(\Delta\varphi_i))$ with respect to the Lebesgue measure on$\mathbb{R}^N$, where $\Delta$ is the discrete Laplacian and we allow for avery large class of potentials $V(\cdot)$. The interaction with the defect lineis introduced by giving the field a reward $\varepsilon\ge0$ each time ittouches the x-axis. We call this model the pinning model. We consider a secondmodel, the wetting model, in which, in addition to the pinning reward, thefield is also constrained to stay nonnegative. We show that both models undergoa phase transition as the intensity $\varepsilon$ of the pinning reward varies:both in the pinning ($a=\mathrm{p}$) and in the wetting ($a=\mathrm{w}$) case,there exists a critical value $\varepsilon_c^a$ such that when$\varepsilon>\varepsilon_c^a$ the field touches the defect line a positivefraction of times (localization), while this does not happen for$\varepsilon<\varepsilon_c^a$ (delocalization). The two critical values arenontrivial and distinct: $0<\varepsilon_c^{\mathrm{p}}<\varepsilon_c^{\mathrm{w}}<\infty$, and they are the onlynonanalyticity points of the respective free energies. For the pinning modelthe transition is of second order, hence the field at$\varepsilon=\varepsilon_c^{\mathrm{p}}$ is delocalized. On the other hand, thetransition in the wetting model is of first order and for$\varepsilon=\varepsilon_c^{\mathrm{w}}$ the field is localized. The core ofour approach is a Markov renewal theory description of the field.

Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction