SCALING LIMITS OF (1+1)-DIMENSIONAL PINNING MODELS WITH LAPLACIAN INTERACTION

• 成果类型: Article
• 署名作者: Caravenna, Francesco; Deuschel, Jean-Dominique
• 署名单位: University of Padua; Technical University of Berlin
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP424
• 发表日期: 2009
• 页码: 903-945
• 关键词: WEAK-CONVERGENCE; inequalities

摘要

We consider a random field phi: {1,..., N} -> R with Laplacian interaction of the form Sigma(i) V(Delta phi(i)), where Delta 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 epsilon >= 0 each time it touches the x-axis, OF that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime (epsilon < epsilon(c)) and a localized one (epsilon > epsilon(c)), where 0 < epsilon(c) < infinity. In this paper we give a precise pathwise description of 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 N(3/2), while in the localized regime the distance is just O((log N)(2)). A subtle scenario shows up in the critical regime (epsilon = epsilon(c)), where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable Levy process of index 2/5. Our approach is based on Markov renewal theory.