Localization for (1+1)- dimensional pinning models with (del plus Delta)- interaction

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 δ-pinning type, with strength ε ≥ 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 ε must be greater than a strictly positive critical threshold εc > 0. On the other hand, when the selfinteraction is purely gradient, it is known that the transition is trivial: an arbitrarily small reward ε > 0 is sufficient to localize the chain at the defect line (εc = 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 εc = 0. © 2010 Applied Probability Trust.