Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction

TL;DR

This paper analyzes the scaling limits of a (1+1)-dimensional Laplacian interaction pinning model, revealing distinct behaviors in localized, delocalized, and critical regimes, with convergence to stable Lévy processes at criticality.

Contribution

It provides a detailed pathwise description of the phase transition, including full scaling limits and the critical behavior involving stable Lévy processes.

Findings

Delocalized regime: field wanders away at scale N^{3/2}

Localized regime: field stays within O((log N)^2) of the defect line

Critical regime: rescaled field converges to the derivative of a stable Lévy process

Abstract

We consider a random field $\varphi:\{1,...,N\}\to \mathbb{R}$ with Laplacian interaction of the form $\sum_iV(\Delta\varphi_i)$, where $\Delta$ is the discrete Laplacian and the potential $V(\cdot)$ is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward $\varepsilon\ge0$ 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 $(\varepsilon<\varepsilon_c)$ and a localized one $(\varepsilon>\varepsilon_c)$, where $0<\varepsilon_c<\infty$. 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 $N^{3/2}$, while in the localized regime the distance is just $O((\log N)^2)$. A subtle scenario shows up in the critical regime $(\varepsilon=\varepsilon_c)$, where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable L\'evy process of index 2/5. Our approach is based on Markov renewal theory.