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May 2009 Scaling limits of (1+1)-dimensional pinning models with Laplacian interaction
Francesco Caravenna, Jean-Dominique Deuschel
Ann. Probab. 37(3): 903-945 (May 2009). DOI: 10.1214/08-AOP424


We consider a random field ϕ:{1, …, N}→ℝ with Laplacian interaction of the form ∑iVϕi), where Δ is the discrete Laplacian and the potential V(⋅) is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward ɛ≥0 each time it touches the x-axis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime (ɛ<ɛc) and a localized one (ɛ>ɛc), where 0<ɛc<∞. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show, in particular, that in the delocalized regime the field wanders away from the defect line at a typical distance N3/2, while in the localized regime the distance is just O((log N)2). A subtle scenario shows up in the critical regime (ɛ=ɛc), where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable Lévy process of index 2/5. Our approach is based on Markov renewal theory.


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Francesco Caravenna. Jean-Dominique Deuschel. "Scaling limits of (1+1)-dimensional pinning models with Laplacian interaction." Ann. Probab. 37 (3) 903 - 945, May 2009.


Published: May 2009
First available in Project Euclid: 19 June 2009

zbMATH: 1185.60106
MathSciNet: MR2537545
Digital Object Identifier: 10.1214/08-AOP424

Primary: 60F05 , 60K35 , 82B41

Keywords: Brascamp–Lieb inequality , Lévy process , Markov renewal theory , phase transition , pinning model , Scaling limit

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 3 • May 2009
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