Pinning and wetting transition for (1+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(Δϕ_i)) with respect to the Lebesgue measure on ℝ^N, where Δ 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 ɛ≥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 ɛ of the pinning reward varies: both in the pinning (a=p) and in the wetting (a=w) case, there exists a critical value ɛ_c^a such that when ɛ>ɛ_c^a the field touches the defect line a positive fraction of times (localization), while this does not happen for ɛ<ɛ_c^a (delocalization). The two critical values are nontrivial and distinct: 0<ɛ_c^p<ɛ_c^w<∞, 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 ɛ=ɛ_c^p is delocalized. On the other hand, the transition in the wetting model is of first order and for ɛ=ɛ_c^w the field is localized. The core of our approach is a Markov renewal theory description of the field.