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(-∑<sub>i</sub> V(Δ φ<sub>i</sub>)) with respect to the Lebesgue measure on R<sup>N</sup>, 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 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 ε<sub>c</sub><sup>a</sup> such that when ε > ε<sub>c</sub><sup>a</sup> the field touches the defect line a positive fraction of times (localization), while this does not happen for ε < ε<sub>c</sub><sup>a</sup> (delocalization). The two critical values are nontrivial and distinct: 0 < ε<sub>c</sub><sup>p</sup> < ε<sub>c</sub><sup>w</sup> < ∞, 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 = ε<sub>c</sub><sup>a</sup> is delocalized. On the other hand, the transition in the wetting model is of first order and for ε = ε<sub>c</sub><sup>a</sup> the field is localized. The core of our approach is a Markov renewal theory description of the field.
Caravenna, F., Deuschel, J. (2008). Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction. ANNALS OF PROBABILITY, 36(6), 2388-2433 [10.1214/08-AOP395].
Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
CARAVENNA, FRANCESCO;
2008
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(-∑i V(Δ φi)) 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 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 εca such 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 = εca is delocalized. On the other hand, the transition in the wetting model is of first order and for ε = εca the field is localized. The core of our approach is a Markov renewal theory description of the field.
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