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

Abstract

We consider a random field $\phi :\{1,...,N\}\to \mathbb{R}$ as a model for a linear chain attracted to the defect line $\phi =0$, that is, the x-axis. The free law of the field is specified by the density $\exp (-{\sum }_{i}V(\Delta {\phi }_i))$ with respect to the Lebesgue measure on ${\mathbb{R}}^N$, where $\Delta$ is the discrete Laplacian and we allow for a very large class of potentials $V(\cdot )$. The interaction with the defect line is introduced by giving the field a reward $\epsilon \ge 0$ each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity $\epsilon$ 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 ${\epsilon }_c^a$ such that when $\epsilon >{\epsilon }_c^a$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $\epsilon <{\epsilon }_c^a$ (delocalization). The two critical values are nontrivial and distinct: $0<\varepsilon_c^{\mat hrm{p}}<\varepsilon_c^{\mathrm{w}}<\infty$, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\epsilon ={\epsilon }_c^{\mathrm{p}}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\epsilon ={\epsilon }_c^{\mathrm{w}}$ the field is localized. The core of our approach is a Markov renewal theory description of the field.

Published in at http://dx.doi.org/10.1214/08-AOP395 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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