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November 2008 Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
Francesco Caravenna, Jean-Dominique Deuschel
Ann. Probab. 36(6): 2388-2433 (November 2008). DOI: 10.1214/08-AOP395


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.


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Francesco Caravenna. Jean-Dominique Deuschel. "Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction." Ann. Probab. 36 (6) 2388 - 2433, November 2008.


Published: November 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1179.60066
MathSciNet: MR2478687
Digital Object Identifier: 10.1214/08-AOP395

Primary: 60F05 , 60K35 , 82B41

Keywords: Entropic repulsion , local limit theorem , Markov renewal theory , Perron–Frobenius theorem , phase transition , pinning model , wetting model

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 6 • November 2008
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