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

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

2008-02-21

arxiv.org/abs/0802.3154v2

Annals of Probability 2009, Vol. 37, No. 3, 903-945

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP424 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/08-AOP424

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.

Affiliated with

Also associated with

No associations

Rating

Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-580105