Random walk versus random line

We consider random walks Xn in Z+, obeying a detailed balance condition, with a weak drift towards the origin when Xn↗∞. We reconsider the equivalence in law between a random walk bridge and a 1+1 dimensional Solid-On-Solid bridge with a corresponding Hamiltonian. Phase diagrams are discussed in terms of recurrence versus wetting. A drift −δXn−1+O(Xn−2) of the random walk yields a Solid-On-Solid potential with an attractive well at the origin and a repulsive tail δ(2+δ)8Xn−2+O(Xn−3) at infinity, showing complete wetting for δ≤1 and critical partial wetting for δ>1.