The Annals of Probability

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

Francesco Caravenna and Jean-Dominique Deuschel

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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(−∑iVϕi)) with respect to the Lebesgue measure on ℝN, 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 ɛca such that when ɛ>ɛca the field touches the defect line a positive fraction of times (localization), while this does not happen for ɛ<ɛca (delocalization). The two critical values are nontrivial and distinct: 0<ɛcp<ɛcw<∞, 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=ɛcp is delocalized. On the other hand, the transition in the wetting model is of first order and for ɛ=ɛcw the field is localized. The core of our approach is a Markov renewal theory description of the field.

Article information

Ann. Probab., Volume 36, Number 6 (2008), 2388-2433.

First available in Project Euclid: 19 December 2008

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F05: Central limit and other weak theorems 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Pinning model wetting model phase transition entropic repulsion Markov renewal theory local limit theorem Perron–Frobenius theorem


Caravenna, Francesco; Deuschel, Jean-Dominique. Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction. Ann. Probab. 36 (2008), no. 6, 2388--2433. doi:10.1214/08-AOP395.

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