We prove an invariance principle for a class of tilted 1 + 1-dimensional SOS models or, equivalently, for a class of tilted random walk bridges in Z_+ . The limiting objects are stationary reversible ergodic diffusions with drifts given by the logarithmic derivatives of the ground states of associated singular Sturm–Liouville operators. In the case of a linear area tilt, we recover the Ferrari–Spohn diffusion with log-Airy drift, which was derived in Ferrari and Spohn (Ann Probab 33(4):1302—1325, 2005) in the context of Brownian motions conditioned to stay above circular and parabolic barriers.
