Localization for (1+1)-dimensional pinning models with $(\nabla + \Delta)$-interaction

TL;DR

This paper investigates a (1+1)-dimensional chain model with mixed gradient and Laplacian interactions, showing that any positive attraction strength leads to localization, indicating a trivial phase transition.

Contribution

It proves that in mixed gradient and Laplacian models, the localization transition occurs at zero attraction strength, extending understanding of phase transitions in such models.

Findings

Transition is trivial: < =0.

Any positive  induces localization.

Results hold under minimal assumptions on potentials.

Abstract

We study the localization/delocalization phase transition in a class of directed models for a homogeneous linear chain attracted to a defect line. The self-interaction of the chain is of mixed gradient and Laplacian kind, whereas the attraction to the defect line is of $\delta$-pinning type, with strength $\epsilon \geq 0$. It is known that, when the self-interaction is purely Laplacian, such models undergo a non-trivial phase transition: to localize the chain at the defect line, the reward $\epsilon$ must be greater than a strictly positive critical threshold $\epsilon_c > 0$. On the other hand, when the self-interaction is purely gradient, it is known that the transition is trivial: an arbitrarily small reward $\epsilon > 0$ is sufficient to localize the chain at the defect line ($\epsilon_c = 0$). In this note we show that in the mixed gradient and Laplacian case, under minimal assumptions on the interaction potentials, the transition is always trivial, that is $\epsilon_c = 0$.