We consider a generalization of the classical pinning problem for integer-valued random walks conditioned to stay non-negative. More specifically, we take pinning potentials of the form $sum_{jgeq 0}epsilon_j N_j$, where $N_j$ is the number of visits to the state $j$ and ${epsilon_j}$ is a non-negative sequence. Partly motivated by similar problems for low-temperature contour models in statistical physics, we aim at finding a sharp characterization of the threshold of the wetting transition, especially in the regime where the variance $sigma^2$ of the single step of the random walk is small. Our main result says that, for natural choices of the pinning sequence ${epsilon_j}$, localization (respectively delocalization) occurs if $sigma^{-2}sum_{ jgeq0}(j+1)epsilon_jgeqdelta^{-1}$ (respectively $le delta$), for some universal $delta 0$ a large enough parameter. This generalization is directly relevant for applications to the above mentioned contour models.
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