  We consider random walks X_n in Z+, obeying a detailed balance condition,
with a weak drift towards the origin when X_n tends to infinity. 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 -delta/X_n of the
random walk yields a Solid-On-Solid potential with an attractive well at the
origin and a repulsive tail delta(delta+2)/(8X_n^2) at infinity, showing
complete wetting for delta<=1 and critical partial wetting for delta>1.
