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

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