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

Abstract

We consider a random field φ :{1,...,N} → ℝ as a model for a linear chain attracted to the defect line φ = 0, that is, the x-axis. The free law of the field is specified by the density exp(- ∑ iV(△φ)) with respect to the Lebesgue measure on ℝN, where △ is the discrete Laplacian and we allow for a very large class of potentials V(.). The interaction with the defect line is introduced by giving the field a reward ε ≥ 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 ε of the pinning reward varies: both in the pinning (a = p) and in the wetting (a =w) case, there exists a critical value asuch that when ε > εca the field touches the defect line a positive fraction of times (localization), while this does not happen for ε > εca (delocalization). The two critical values are nontrivial and distinct: 0 > εcp > εcw > ∞, 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 ε = εcp is delocalized. On the other hand, the transition in the wetting model is of first order and for e = εcw the field is localized. The core of our approach is a Markov renewal theory description of the field. © Institute of Mathematical Statistics. 2008.