Concentration under scaling limits for weakly pinned Gaussian random walks

We study scaling limits for d-dimensional Gaussian random walks perturbed by an attractive force toward a certain subspace of R^d, especially under the critical situation that the rate functional of the corresponding large deviation principle admits two minimizers. We obtain different type of limits, in a positive recurrent regime, depending on the co-dimension of the subspace and the conditions imposed at the final time under the presence or absence of a wall. The motivation comes from the study of polymers or (1 + 1)-dimensional interfaces with δ-pinning.